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134-545: The PSLRA was designed to limit frivolous securities lawsuits . Prior to the PSLRA, plaintiffs could proceed with minimal evidence of fraud, and then use pretrial discovery to seek further proof. That set a very low barrier to initiate litigation, which encouraged the filing of weak or entirely-frivolous suits. Defending against the suits could prove extremely costly, even when the charges were unfounded, and so defendants often found it cheaper to settle than to fight and win. Under

268-521: A b x f ( x ) d x = ∫ a b x x 2 + π 2 d x = 1 2 ln ⁡ b 2 + π 2 a 2 + π 2 . {\displaystyle \int _{a}^{b}xf(x)\,dx=\int _{a}^{b}{\frac {x}{x^{2}+\pi ^{2}}}\,dx={\frac {1}{2}}\ln {\frac {b^{2}+\pi ^{2}}{a^{2}+\pi ^{2}}}.} The limit of this expression as

402-437: A broker-dealer who trades with other broker-dealers, rather than with the retail investor. This distinction carries over to banking ; compare Retail banking and Wholesale banking . The traditional economic function of the purchase of securities is investment, with the view to receiving income or achieving capital gain . Debt securities generally offer a higher rate of interest than bank deposits, and equities may offer

536-471: A motion to dismiss under Rule 12(b)(6) of the Federal Rules of Civil Procedure . A motion to dismiss under Rule 12(b)(6) is essentially an argument by the defendant that even if all of the facts alleged in the complaint were assumed to be true, they would not be sufficient to give rise to liability under Rule 10b-5. If the court determines that the facts alleged in the complaint are sufficient to state

670-401: A weighted average of the x i values, with weights given by their probabilities p i . In the special case that all possible outcomes are equiprobable (that is, p 1 = ⋅⋅⋅ = p k ), the weighted average is given by the standard average . In the general case, the expected value takes into account the fact that some outcomes are more likely than others. Informally,

804-421: A → −∞ and b → ∞ does not exist: if the limits are taken so that a = − b , then the limit is zero, while if the constraint 2 a = − b is taken, then the limit is ln(2) . To avoid such ambiguities, in mathematical textbooks it is common to require that the given integral converges absolutely , with E[ X ] left undefined otherwise. However, measure-theoretic notions as given below can be used to give

938-404: A 10b-5 case. The stock analyst cases provide a good example of the importance of how the "relevant truth" is defined. The claim in most stock analyst cases is that the analyst dishonestly touted a particular stock as a good investment despite the analyst's genuine belief. Under Dura , such cases are often dismissed because, in most instances, the analyst's dishonesty never comes to light until after

1072-408: A Rule 10b-5 action: Each of the elements has been heavily litigated in thousands of cases over the past 70 years, and the courts have applied the six elements in a multitude of different factual situations. In a typical Rule 10b-5 claim, the plaintiff starts the action by filing a complaint in federal court . Other actions alleging substantially similar claims may also be filed. The PSLRA mandates

1206-508: A Rule 10b-5 claim, the plaintiff then becomes entitled to obtain "discovery" from the defendant, which typically means the right to demand documentary evidence in the defendant's possession concerning the facts at issue, and the right to require the defendant (and other witnesses) to sit for depositions. In a modern litigation context, the discovery process is protracted and very expensive and often involves production of millions of pages of documents and depositions of dozens of witnesses and costs

1340-647: A bill into law. The PSLRA was originally developed as part of Newt Gingrich 's Contract With America . Its principal authors in the House were Representatives Thomas Bliley , Jack Fields and Christopher Cox . Senators Chris Dodd and Pete Domenici sponsored the legislation in the Senate. Security (finance) A security is a tradable financial asset . The term commonly refers to any form of financial instrument , but its legal definition varies by jurisdiction. In some countries and languages people commonly use

1474-680: A certificate or, more typically, they may be "non-certificated", that is in electronic ( dematerialized ) or " book entry only" form. Certificates may be bearer , meaning they entitle the holder to rights under the security merely by holding the security, or registered , meaning they entitle the holder to rights only if they appear on a security register maintained by the issuer or an intermediary. They include shares of corporate capital stock or mutual funds , bonds issued by corporations or governmental agencies, stock options or other options, limited partnership units, and various other formal investment instruments that are negotiable and fungible . In

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1608-420: A combination of the two is used. The distinction between the two is important to securities regulation and company law . Privately placed securities are not publicly tradable and may only be bought and sold by sophisticated qualified investors. As a result, the secondary market is not nearly as liquid as it is for public (registered) securities. Another category, sovereign bonds , is generally sold by auction to

1742-491: A debt security is typically entitled to the payment of principal and interest, together with other contractual rights under the terms of the issue, such as the right to receive certain information. Debt securities are generally issued for a fixed term and redeemable by the issuer at the end of that term. Debt securities may be protected by collateral or may be unsecured, and, if they are unsecured, may be contractually "senior" to other unsecured debt meaning their holders would have

1876-512: A finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration . In the axiomatic foundation for probability provided by measure theory , the expectation is given by Lebesgue integration . The expected value of a random variable X is often denoted by E( X ) , E[ X ] , or E X , with E also often stylized as E {\displaystyle \mathbb {E} } or E . The idea of

2010-542: A lower rate of interest than corporate bonds, and serve as a source of finance for governments. U.S. federal government bonds are called treasuries. Because of their liquidity and perceived low risk, treasuries are used to manage the money supply in the open market operations of non-US central banks. Sub-sovereign government bonds , known in the U.S. as municipal bonds , represent the debt of state, provincial, territorial, municipal or other governmental units other than sovereign governments. Supranational bonds represent

2144-488: A multidimensional random variable, i.e. a random vector X . It is defined component by component, as E[ X ] i = E[ X i ] . Similarly, one may define the expected value of a random matrix X with components X ij by E[ X ] ij = E[ X ij ] . Consider a random variable X with a finite list x 1 , ..., x k of possible outcomes, each of which (respectively) has probability p 1 , ..., p k of occurring. The expectation of X

2278-410: A plaintiff in a Rule 10b-5 case had not adequately pleaded loss causation by merely alleging that he "paid artificially inflated prices for Dura securities" at the time of purchase. The Supreme Court observed that an investor who purchases a stock at an artificially-inflated price suffers no economic loss at the time of purchase. The loss occurs only when the truth is disclosed and the stock price falls as

2412-707: A priority in a bankruptcy of the issuer. Debt that is not senior is "subordinated". Corporate bonds represent the debt of commercial or industrial entities. Debentures have a long maturity, typically at least ten years, whereas notes have a shorter maturity. Commercial paper is a simple form of debt security that essentially represents a post-dated cheque with a maturity of not more than 270 days. Money market instruments are short term debt instruments that may have characteristics of deposit accounts, such as certificates of deposit , Accelerated Return Notes (ARN) , and certain bills of exchange . They are highly liquid and are sometimes referred to as "near cash". Commercial paper

2546-570: A real number μ {\displaystyle \mu } if and only if the two surfaces in the x {\displaystyle x} - y {\displaystyle y} -plane, described by x ≤ μ , 0 ≤ y ≤ F ( x ) or x ≥ μ , F ( x ) ≤ y ≤ 1 {\displaystyle x\leq \mu ,\;\,0\leq y\leq F(x)\quad {\text{or}}\quad x\geq \mu ,\;\,F(x)\leq y\leq 1} respectively, have

2680-411: A register in which details of the holder of the securities are entered and updated as appropriate. A transfer of registered securities is effected by amending the register. Modern practice has developed to eliminate both the need for certificates and maintenance of a complete security register by the issuer. There are two general ways this has been accomplished. In some jurisdictions, such as France, it

2814-462: A result. Thus, a plaintiff who sells his shares "before the relevant truth begins to leak out" does not suffer any economic damage. The plaintiff in Dura failed to allege that the "share price fell significantly after the truth became known" and so the complaint had not alleged loss causation. How a court defines the "relevant truth" that the defendant fraudulently withheld often determines the outcome of

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2948-399: A round-table of market data industry firms, referring to them as Consumers, Exchanges, and Vendors. In India the equivalent organisation is the securities exchange board of India (SEBI). In the primary markets, securities may be offered to the public in a public offering . Alternatively, they may be offered privately to a limited number of qualified persons in a private placement . Sometimes

3082-456: A shelf registration. These later new issues are also sold in the primary market, but they are not considered to be an IPO but are often called a "secondary offering". Issuers usually retain investment banks to assist them in administering the IPO, obtaining SEC (or other regulatory body) approval of the offering filing, and selling the new issue. When the investment bank buys the entire new issue from

3216-459: A small circle of mutual scientific friends in Paris about it. In Dutch mathematician Christiaan Huygens' book, he considered the problem of points, and presented a solution based on the same principle as the solutions of Pascal and Fermat. Huygens published his treatise in 1657, (see Huygens (1657) ) " De ratiociniis in ludo aleæ " on probability theory just after visiting Paris. The book extended

3350-404: A specialized class of dealers. Securities are often listed in a stock exchange , an organized and officially recognized market on which securities can be bought and sold. Issuers may seek listings for their securities to attract investors, by ensuring there is a liquid and regulated market that investors can buy and sell securities in. Growth in informal electronic trading systems has challenged

3484-616: A systematic definition of E[ X ] for more general random variables X . All definitions of the expected value may be expressed in the language of measure theory . In general, if X is a real-valued random variable defined on a probability space (Ω, Σ, P) , then the expected value of X , denoted by E[ X ] , is defined as the Lebesgue integral E ⁡ [ X ] = ∫ Ω X d P . {\displaystyle \operatorname {E} [X]=\int _{\Omega }X\,d\operatorname {P} .} Despite

3618-507: A value in any given open interval is given by the integral of f over that interval. The expectation of X is then given by the integral E ⁡ [ X ] = ∫ − ∞ ∞ x f ( x ) d x . {\displaystyle \operatorname {E} [X]=\int _{-\infty }^{\infty }xf(x)\,dx.} A general and mathematically precise formulation of this definition uses measure theory and Lebesgue integration , and

3752-469: A variety of stylizations: the expectation operator can be stylized as E (upright), E (italic), or E {\displaystyle \mathbb {E} } (in blackboard bold ), while a variety of bracket notations (such as E( X ) , E[ X ] , and E X ) are all used. Another popular notation is μ X . ⟨ X ⟩ , ⟨ X ⟩ av , and X ¯ {\displaystyle {\overline {X}}} are commonly used in physics. M( X )

3886-640: Is a Borel function ), we can use this inversion formula to obtain E ⁡ [ g ( X ) ] = 1 2 π ∫ R g ( x ) [ ∫ R e − i t x φ X ( t ) d t ] d x . {\displaystyle \operatorname {E} [g(X)]={\frac {1}{2\pi }}\int _{\mathbb {R} }g(x)\left[\int _{\mathbb {R} }e^{-itx}\varphi _{X}(t)\,dt\right]dx.} If E ⁡ [ g ( X ) ] {\displaystyle \operatorname {E} [g(X)]}

4020-415: Is a generalization of the weighted average . Informally, the expected value is the mean of the possible values a random variable can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would "expect" to get in reality. The expected value of a random variable with

4154-402: Is a significant change in an issuer's stock price, without regard to any underlying culpability of the issuer, and with only faint hope that the discovery process might lead eventually to some plausible cause of action." To reduce the number of purportedly-frivolous Rule 10b-5 lawsuits that survive motions to dismiss, the PSLRA raised the pleading standards for the specificity and the strength of

Private Securities Litigation Reform Act - Misplaced Pages Continue

4288-426: Is also a form of capital stock. The holder of an equity is a shareholder, owning a share, or fractional part of the issuer. Unlike debt securities, which typically require regular payments (interest) to the holder, equity securities are not entitled to any payment. In bankruptcy, they share only in the residual interest of the issuer after all obligations have been paid out to creditors. However, equity generally entitles

4422-541: Is also often highly liquid. Euro debt securities are securities issued internationally outside their domestic market in a denomination different from that of the issuer's domicile. They include eurobonds and euronotes. Eurobonds are characteristically underwritten, and not secured, and interest is paid gross. A euronote may take the form of euro-commercial paper (ECP) or euro-certificates of deposit. Government bonds are medium or long term debt securities issued by sovereign governments or their agencies. Typically they carry

4556-546: Is any random variable with finite expectation, then Markov's inequality may be applied to the random variable | X −E[ X ]| to obtain Chebyshev's inequality P ⁡ ( | X − E [ X ] | ≥ a ) ≤ Var ⁡ [ X ] a 2 , {\displaystyle \operatorname {P} (|X-{\text{E}}[X]|\geq a)\leq {\frac {\operatorname {Var} [X]}{a^{2}}},} where Var

4690-462: Is as in the previous example. A number of convergence results specify exact conditions which allow one to interchange limits and expectations, as specified below. The probability density function f X {\displaystyle f_{X}} of a scalar random variable X {\displaystyle X} is related to its characteristic function φ X {\displaystyle \varphi _{X}} by

4824-635: Is called the probability density function of X (relative to Lebesgue measure). According to the change-of-variables formula for Lebesgue integration, combined with the law of the unconscious statistician , it follows that E ⁡ [ X ] ≡ ∫ Ω X d P = ∫ R x f ( x ) d x {\displaystyle \operatorname {E} [X]\equiv \int _{\Omega }X\,d\operatorname {P} =\int _{\mathbb {R} }xf(x)\,dx} for any absolutely continuous random variable X . The above discussion of continuous random variables

4958-408: Is defined as E ⁡ [ X ] = x 1 p 1 + x 2 p 2 + ⋯ + x k p k . {\displaystyle \operatorname {E} [X]=x_{1}p_{1}+x_{2}p_{2}+\cdots +x_{k}p_{k}.} Since the probabilities must satisfy p 1 + ⋅⋅⋅ + p k = 1 , it is natural to interpret E[ X ] as

5092-407: Is easily obtained by setting Y 0 = X 1 {\displaystyle Y_{0}=X_{1}} and Y n = X n + 1 − X n {\displaystyle Y_{n}=X_{n+1}-X_{n}} for n ≥ 1 , {\displaystyle n\geq 1,} where X n {\displaystyle X_{n}}

5226-564: Is equivalent to the representation E ⁡ [ X ] = ∫ 0 ∞ ( 1 − F ( x ) ) d x − ∫ − ∞ 0 F ( x ) d x , {\displaystyle \operatorname {E} [X]=\int _{0}^{\infty }{\bigl (}1-F(x){\bigr )}\,dx-\int _{-\infty }^{0}F(x)\,dx,} also with convergent integrals. Expected values as defined above are automatically finite numbers. However, in many cases it

5360-436: Is finite if and only if E[ X ] and E[ X ] are both finite. Due to the formula | X | = X + X , this is the case if and only if E| X | is finite, and this is equivalent to the absolute convergence conditions in the definitions above. As such, the present considerations do not define finite expected values in any cases not previously considered; they are only useful for infinite expectations. The following table gives

5494-626: Is finite, changing the order of integration, we get, in accordance with Fubini–Tonelli theorem , E ⁡ [ g ( X ) ] = 1 2 π ∫ R G ( t ) φ X ( t ) d t , {\displaystyle \operatorname {E} [g(X)]={\frac {1}{2\pi }}\int _{\mathbb {R} }G(t)\varphi _{X}(t)\,dt,} where G ( t ) = ∫ R g ( x ) e − i t x d x {\displaystyle G(t)=\int _{\mathbb {R} }g(x)e^{-itx}\,dx}

Private Securities Litigation Reform Act - Misplaced Pages Continue

5628-1055: Is fundamental to be able to consider expected values of ±∞ . This is intuitive, for example, in the case of the St. Petersburg paradox , in which one considers a random variable with possible outcomes x i = 2 , with associated probabilities p i = 2 , for i ranging over all positive integers. According to the summation formula in the case of random variables with countably many outcomes, one has E ⁡ [ X ] = ∑ i = 1 ∞ x i p i = 2 ⋅ 1 2 + 4 ⋅ 1 4 + 8 ⋅ 1 8 + 16 ⋅ 1 16 + ⋯ = 1 + 1 + 1 + 1 + ⋯ . {\displaystyle \operatorname {E} [X]=\sum _{i=1}^{\infty }x_{i}\,p_{i}=2\cdot {\frac {1}{2}}+4\cdot {\frac {1}{4}}+8\cdot {\frac {1}{8}}+16\cdot {\frac {1}{16}}+\cdots =1+1+1+1+\cdots .} It

5762-560: Is growing slowly. Securities that are represented in paper (physical) form are called certificated securities. They may be bearer or registered . Securities may also be held in the Direct Registration System (DRS), which is a method of recording shares of stock in book-entry form. Book-entry means the company's transfer agent maintains the shares on the owner's behalf without the need for physical share certificates. Shares held in un-certificated book-entry form have

5896-455: Is multiplied by the small chance of losing a jury verdict, may still be very high. Thus, the outcome of a motion to dismiss in a Rule 10b-5 case essentially determines whether the case gets dismissed, or it proceeds to discovery and very often results in class certification and a very large settlement. Because the motion to dismiss is a pivotal stage in the course of a Rule 10b-5 case, the attorneys for plaintiffs and defendants argue fiercely over

6030-472: Is natural to say that the expected value equals +∞ . There is a rigorous mathematical theory underlying such ideas, which is often taken as part of the definition of the Lebesgue integral. The first fundamental observation is that, whichever of the above definitions are followed, any nonnegative random variable whatsoever can be given an unambiguous expected value; whenever absolute convergence fails, then

6164-642: Is otherwise available. For example, in the case of an unweighted dice, Chebyshev's inequality says that odds of rolling between 1 and 6 is at least 53%; in reality, the odds are of course 100%. The Kolmogorov inequality extends the Chebyshev inequality to the context of sums of random variables. The following three inequalities are of fundamental importance in the field of mathematical analysis and its applications to probability theory. The Hölder and Minkowski inequalities can be extended to general measure spaces , and are often given in that context. By contrast,

6298-603: Is possible for issuers of that jurisdiction to maintain a legal record of their securities electronically. In the United States , the current "official" version of Article 8 of the Uniform Commercial Code permits non-certificated securities. However, the "official" UCC is a mere draft that must be enacted individually by each U.S. state . Though all 50 states (as well as the District of Columbia and

6432-432: Is provided by investors who purchase the securities upon their initial issuance. In a similar way, a government may issue securities when it chooses to increase government debt . Securities are traditionally divided into debt securities and equities. Debt securities may be called debentures , bonds , deposits , notes or commercial paper depending on their maturity, collateral and other characteristics. The holder of

6566-604: Is the Fourier transform of g ( x ) . {\displaystyle g(x).} The expression for E ⁡ [ g ( X ) ] {\displaystyle \operatorname {E} [g(X)]} also follows directly from the Plancherel theorem . The expectation of a random variable plays an important role in a variety of contexts. In statistics , where one seeks estimates for unknown parameters based on available data gained from samples ,

6700-482: Is the variance . These inequalities are significant for their nearly complete lack of conditional assumptions. For example, for any random variable with finite expectation, the Chebyshev inequality implies that there is at least a 75% probability of an outcome being within two standard deviations of the expected value. However, in special cases the Markov and Chebyshev inequalities often give much weaker information than

6834-682: Is the "most plausible," it is not "strong" and the complaint should be dismissed. Two circuits, the Eighth and the Tenth, consider all inferences, both of scienter and of an innocent mental state, using the innocent inferences to test whether the culpable interest is strong, but do not directly weigh one against the other. The Second and Third circuits divide the factual allegations bearing on a defendant's mental state into categories, "motive and opportunity" and "strong circumstantial evidence" of knowing or reckless conduct, either of which may independently satisfy

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6968-747: Is the centre of the eurosecurities markets. There was a huge rise in the eurosecurities market in London in the early 1980s. Settlement of trades in eurosecurities is currently effected through two European computerized clearing/depositories called Euroclear (in Belgium) and Clearstream (formerly Cedelbank) in Luxembourg. The main market for Eurobonds is the EuroMTS, owned by Borsa Italiana and Euronext. There are ramp up market in Emergent countries, but it

7102-1673: Is then natural to define: E ⁡ [ X ] = { E ⁡ [ X + ] − E ⁡ [ X − ] if  E ⁡ [ X + ] < ∞  and  E ⁡ [ X − ] < ∞ ; + ∞ if  E ⁡ [ X + ] = ∞  and  E ⁡ [ X − ] < ∞ ; − ∞ if  E ⁡ [ X + ] < ∞  and  E ⁡ [ X − ] = ∞ ; undefined if  E ⁡ [ X + ] = ∞  and  E ⁡ [ X − ] = ∞ . {\displaystyle \operatorname {E} [X]={\begin{cases}\operatorname {E} [X^{+}]-\operatorname {E} [X^{-}]&{\text{if }}\operatorname {E} [X^{+}]<\infty {\text{ and }}\operatorname {E} [X^{-}]<\infty ;\\+\infty &{\text{if }}\operatorname {E} [X^{+}]=\infty {\text{ and }}\operatorname {E} [X^{-}]<\infty ;\\-\infty &{\text{if }}\operatorname {E} [X^{+}]<\infty {\text{ and }}\operatorname {E} [X^{-}]=\infty ;\\{\text{undefined}}&{\text{if }}\operatorname {E} [X^{+}]=\infty {\text{ and }}\operatorname {E} [X^{-}]=\infty .\end{cases}}} According to this definition, E[ X ] exists and

7236-514: Is thus a special case of the general Lebesgue theory, due to the fact that every piecewise-continuous function is measurable. The expected value of any real-valued random variable X {\displaystyle X} can also be defined on the graph of its cumulative distribution function F {\displaystyle F} by a nearby equality of areas. In fact, E ⁡ [ X ] = μ {\displaystyle \operatorname {E} [X]=\mu } with

7370-466: Is used in Russian-language literature. As discussed above, there are several context-dependent ways of defining the expected value. The simplest and original definition deals with the case of finitely many possible outcomes, such as in the flip of a coin. With the theory of infinite series, this can be extended to the case of countably many possible outcomes. It is also very common to consider

7504-403: Is worth just such a Sum, as wou'd procure in the same Chance and Expectation at a fair Lay. ... If I expect a or b, and have an equal chance of gaining them, my Expectation is worth (a+b)/2. More than a hundred years later, in 1814, Pierre-Simon Laplace published his tract " Théorie analytique des probabilités ", where the concept of expected value was defined explicitly: ... this advantage in

7638-647: The Luxembourg Stock Exchange or admitted to listing in London . The reasons for listing eurobonds include regulatory and tax considerations, as well as the investment restrictions. Securities Services refers to the products and services that are offered to institutional clients that issue, trade, and hold securities. The bank engaged in securities services are usually called a custodian bank. Market players include BNY Mellon , J.P. Morgan , HSBC , Citi , BNP Paribas , Société Générale etc. London

7772-461: The U.S. Virgin Islands ) have enacted some form of Article 8, many of them still appear to use older versions of Article 8, including some that did not permit non-certificated securities. Expected value In probability theory , the expected value (also called expectation , expectancy , expectation operator , mathematical expectation , mean , expectation value , or first moment )

7906-855: The principal trade organization for securities dealers is the International Capital Market Association. In the U.S., the principal trade organization for securities dealers is the Securities Industry and Financial Markets Association, which is the result of the merger of the Securities Industry Association and the Bond Market Association. The Financial Information Services Division of the Software and Information Industry Association (FISD/SIIA) represents

8040-408: The sample mean serves as an estimate for the expectation, and is itself a random variable. In such settings, the sample mean is considered to meet the desirable criterion for a "good" estimator in being unbiased ; that is, the expected value of the estimate is equal to the true value of the underlying parameter. For a different example, in decision theory , an agent making an optimal choice in

8174-478: The "upside" of the business and to control the business. Hybrid securities combine some of the characteristics of both debt and equity securities. Preference shares form an intermediate class of security between equities and debt. If the issuer is liquidated, preference shareholders have the right to receive interest or a return of capital prior to ordinary shareholders. However, from a legal perspective, preference shares are capital stocks and therefore may entitle

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8308-535: The Democratic Party like Senators Tom Harkin , Ted Kennedy , Claiborne Pell , and Carol Moseley Braun voted in favor of the legislation, but many conservative-to-moderate Democrats such as Senators John Breaux , Robert Byrd , Fritz Hollings , and Sam Nunn and Representatives such as John Murtha and Gene Taylor voted against it. There was one other time during Bill Clinton's presidency that Congress successfully overrode one of his 37 vetoes to enact

8442-530: The Exchange Act (codified at 15 U.S.C. § 78j ), as well as pursuant to SEC Rule 10b-5 , which the SEC promulgated under the authority granted to it by Congress under the Exchange Act. Federal securities fraud actions will be referred to as "Rule 10b-5 actions" or "Rule 10b-5 cases" as convenient shorthand. The Supreme Court has held that there are six elements that a plaintiff must allege and prove to prevail in

8576-550: The Federal Rules of Civil Procedure. If the court grants the plaintiff's motion for class certification, the case becomes a securities fraud class action, and the defendant then usually faces enormous liability if the case goes to trial, and the jury renders a verdict against the defendant. For example, if the plaintiff class is composed of 50,000 shareholders, each shareholder owns an average of 1,000 shares, and each shareholder claims to have sustained losses of $ 10 per share,

8710-531: The Jensen inequality is special to the case of probability spaces. In general, it is not the case that E ⁡ [ X n ] → E ⁡ [ X ] {\displaystyle \operatorname {E} [X_{n}]\to \operatorname {E} [X]} even if X n → X {\displaystyle X_{n}\to X} pointwise. Thus, one cannot interchange limits and expectation, without additional conditions on

8844-398: The Lebesgue theory of expectation is identical to the summation formulas given above. However, the Lebesgue theory clarifies the scope of the theory of probability density functions. A random variable X is said to be absolutely continuous if any of the following conditions are satisfied: These conditions are all equivalent, although this is nontrivial to establish. In this definition, f

8978-600: The Official List. In the United States, a "security" is a tradable financial asset of any kind. Securities can be broadly categorized into: The company or other entity issuing the security is called the issuer . A country's regulatory structure determines what qualifies as a security. For example, private investment pools may have some features of securities, but they may not be registered or regulated as such if they meet various restrictions. Securities are

9112-452: The PSLRA, however, plaintiffs must meet a heightened pleading standard before they can initiate a suit. In other words, PSLRA was specifically intended to make it more difficult to initiate securities litigation (frivolous or otherwise) because plaintiffs would supposedly be forced to present evidence of fraud before any pretrial discovery has taken place. Also, PSLRA was enacted to "give teeth to Section 11 ." Congress specifically pointed to

9246-635: The United Kingdom, the Financial Conduct Authority functions as the national competent authority for the regulation of financial markets; the definition in its Handbook of the term "security" applies only to equities, debentures , alternative debentures, government and public securities, warrants, certificates representing certain securities, units, stakeholder pension schemes, personal pension schemes, rights to or interests in investments, and anything that may be admitted to

9380-470: The amount of attorneys' fees . It bars bonus payments to favored plaintiffs and permits judges to scrutinize lawyer conflicts of interest . The Securities Exchange Act of 1934 (commonly known as the "Exchange Act" or the "1934 Act") gives shareholders the right to bring a private action in federal court to recover damages the shareholder sustained as a result of securities fraud. The majority of securities fraud claims are brought pursuant to Section 10(b) of

9514-411: The challenged statement was not in fact misleading. Rule 9(b) of the Federal Rules of Civil Procedure had already required for allegations of fraud to be pleaded with particularity. The PSLRA also requires a plaintiff to allege that the defendant acted with the required state of mind and knew the challenged statement was false at the time it was made or was reckless in not recognizing that the statement

9648-481: The company will call the bond by giving the holder the call price, which may be less than the value of the converted stock. This is referred to as a forced conversion. Equity warrants are options issued by the company that allow the holder of the warrant to purchase a specific number of shares at a specified price within a specified time. They are often issued together with bonds or existing equities, and are, sometimes, detachable from them and separately tradeable. When

9782-437: The complaint shall state with particularity all facts on which that belief is formed." 15 U.S.C. § 78u-4(b)(1). If a plaintiff's complaint does not specifically identify the allegedly fraudulent statements and explain why they were misleading, the complaint is dismissed. By requiring plaintiffs to set forth their theory regarding why a particular statement was misleading, the PSLRA enables defendants to put forth arguments as to why

9916-425: The compulsory deposit and immobilization of bearer shares and units with a depositary allowing identification of the holders thereof. In the case of registered securities, certificates bearing the name of the holder are issued, but these merely represent the securities. A person does not automatically acquire legal ownership by having possession of the certificate. Instead, the issuer (or its appointed agent) maintains

10050-507: The concept of expectation by adding rules for how to calculate expectations in more complicated situations than the original problem (e.g., for three or more players), and can be seen as the first successful attempt at laying down the foundations of the theory of probability . In the foreword to his treatise, Huygens wrote: It should be said, also, that for some time some of the best mathematicians of France have occupied themselves with this kind of calculus so that no one should attribute to me

10184-430: The consolidation of those actions into one lawsuit followed by the appointment of a plaintiff or group of plaintiffs to serve as the lead plaintiff. More than one plaintiff typically seeks to serve as the lead plaintiff. The court overseeing the lawsuit typically appoints the plaintiff with the "largest financial interest" in the action, or greatest sustained loss, to serve as the plead plaintiff. The defendant will then file

10318-459: The consumer level, loans against securities have grown into three distinct groups over the last decade: Of the three, transfer-of-title loans have fallen into the very high-risk category as the number of providers has dwindled as regulators have launched an industry-wide crackdown on transfer-of-title structures where the private lender may sell or sell short the securities to fund the loan. Institutionally managed consumer securities-based loans on

10452-462: The corresponding theory of absolutely continuous random variables is described in the next section. The density functions of many common distributions are piecewise continuous , and as such the theory is often developed in this restricted setting. For such functions, it is sufficient to only consider the standard Riemann integration . Sometimes continuous random variables are defined as those corresponding to this special class of densities, although

10586-725: The debt of international organizations such as the World Bank , the International Monetary Fund , regional multilateral development banks like the African Development Bank and the Asian Development Bank , and others. An equity security is a share of equity interest in an entity such as the capital stock of a company, trust or partnership. The most common form of equity interest is common stock, although preferred equity

10720-408: The defendant as much as millions of dollars in legal fees. Thus, it is extremely important to the litigants if a motion to dismiss is granted, as the costs of litigation increase substantially when the motion to dismiss is denied. If the plaintiff's complaint survives the defendant's Rule 12(b)(6) motion to dismiss, the next step is usually for the plaintiff to seek class certification under Rule 23 of

10854-490: The distinct case of random variables dictated by (piecewise-)continuous probability density functions , as these arise in many natural contexts. All of these specific definitions may be viewed as special cases of the general definition based upon the mathematical tools of measure theory and Lebesgue integration , which provide these different contexts with an axiomatic foundation and common language. Any definition of expected value may be extended to define an expected value of

10988-598: The evasion of regulatory restrictions and tax. In the United Kingdom , for example, the issue of bearer securities was heavily restricted firstly by the Exchange Control Act 1947 until 1953. Bearer securities are very rare in the United States because of the negative tax implications they may have to the issuer and holder. In Luxembourg, the law of 28 July 2014 concerning the compulsory deposit and immobilization of shares and units in bearer form adopts

11122-527: The expectation of a random variable with a countably infinite set of possible outcomes is defined analogously as the weighted average of all possible outcomes, where the weights are given by the probabilities of realizing each given value. This is to say that E ⁡ [ X ] = ∑ i = 1 ∞ x i p i , {\displaystyle \operatorname {E} [X]=\sum _{i=1}^{\infty }x_{i}\,p_{i},} where x 1 , x 2 , ... are

11256-492: The expected value can be defined as +∞ . The second fundamental observation is that any random variable can be written as the difference of two nonnegative random variables. Given a random variable X , one defines the positive and negative parts by X = max( X , 0) and X = −min( X , 0) . These are nonnegative random variables, and it can be directly checked that X = X − X . Since E[ X ] and E[ X ] are both then defined as either nonnegative numbers or +∞ , it

11390-494: The expected value operator is not σ {\displaystyle \sigma } -additive, i.e. E ⁡ [ ∑ n = 0 ∞ Y n ] ≠ ∑ n = 0 ∞ E ⁡ [ Y n ] . {\displaystyle \operatorname {E} \left[\sum _{n=0}^{\infty }Y_{n}\right]\neq \sum _{n=0}^{\infty }\operatorname {E} [Y_{n}].} An example

11524-573: The expected value originated in the middle of the 17th century from the study of the so-called problem of points , which seeks to divide the stakes in a fair way between two players, who have to end their game before it is properly finished. This problem had been debated for centuries. Many conflicting proposals and solutions had been suggested over the years when it was posed to Blaise Pascal by French writer and amateur mathematician Chevalier de Méré in 1654. Méré claimed that this problem could not be solved and that it showed just how flawed mathematics

11658-488: The expected values of some commonly occurring probability distributions . The third column gives the expected values both in the form immediately given by the definition, as well as in the simplified form obtained by computation therefrom. The details of these computations, which are not always straightforward, can be found in the indicated references. The basic properties below (and their names in bold) replicate or follow immediately from those of Lebesgue integral . Note that

11792-422: The factual allegations that must be alleged in the plaintiff's complaint in three specific ways. Those changes are among the most significant aspects of the PSLRA. The PSLRA requires a plaintiff to identify in his complaint "each statement alleged to have been misleading, the reason or reasons why the statement is misleading, and if an allegation regarding the statement or omission is made on information and belief,

11926-511: The firm's Board of Directors, or the Board of Directors' Audit Committee. The Board of Directors or subunit is consequently obligated to notify the Securities and Exchange Commission of such notification by auditors within one day. If the auditing accountant does not receive a copy of the SEC notification within one day, the auditor must withdraw from the engagement and notify the SEC directly of

12060-524: The holder of the warrant exercises it, he pays the money directly to the company, and the company issues new shares to the holder. Warrants, like other convertible securities, increases the number of shares outstanding, and are always accounted for in financial reports as fully diluted earnings per share, which assumes that all warrants and convertibles will be exercised. Securities may be classified according to many categories or classification systems: Investors in securities may be retail , i.e., members of

12194-481: The holder to a pro rata portion of control of the company, meaning that a holder of a majority of the equity is usually entitled to control the issuer. Equity also enjoys the right to profits and capital gain , whereas holders of debt securities receive only interest and repayment of principal regardless of how well the issuer performs financially. Furthermore, debt securities do not have voting rights outside of bankruptcy. In other words, equity holders are entitled to

12328-424: The holders to some degree of control depending on whether they carry voting rights. Convertibles are bonds or preferred stocks that can be converted, at the election of the holder of the convertibles, into the ordinary shares of the issuing company. The convertibility, however, may be forced if the convertible is a callable bond , and the issuer calls the bond. The bondholder has about one month to convert it, or

12462-401: The honour of the first invention. This does not belong to me. But these savants, although they put each other to the test by proposing to each other many questions difficult to solve, have hidden their methods. I have had therefore to examine and go deeply for myself into this matter by beginning with the elements, and it is impossible for me for this reason to affirm that I have even started from

12596-1193: The indicator function of the event A . {\displaystyle A.} Then, it follows that X n → 0 {\displaystyle X_{n}\to 0} pointwise. But, E ⁡ [ X n ] = n ⋅ Pr ( U ∈ [ 0 , 1 n ] ) = n ⋅ 1 n = 1 {\displaystyle \operatorname {E} [X_{n}]=n\cdot \Pr \left(U\in \left[0,{\tfrac {1}{n}}\right]\right)=n\cdot {\tfrac {1}{n}}=1} for each n . {\displaystyle n.} Hence, lim n → ∞ E ⁡ [ X n ] = 1 ≠ 0 = E ⁡ [ lim n → ∞ X n ] . {\displaystyle \lim _{n\to \infty }\operatorname {E} [X_{n}]=1\neq 0=\operatorname {E} \left[\lim _{n\to \infty }X_{n}\right].} Analogously, for general sequence of random variables { Y n : n ≥ 0 } , {\displaystyle \{Y_{n}:n\geq 0\},}

12730-454: The inference must be "cogent" and "at least as compelling as any opposing inference of nonfraudulent intent." The US Circuit Courts of Appeal had adopted substantially-disparate tests for determining whether scienter is sufficiently pleaded: Four circuits, the First, Fourth, Sixth and Ninth, require a direct comparison of the plausibility of competing inferences. Unless the culpable inference

12864-399: The infinite sum is a finite number independent of the ordering of summands. In the alternative case that the infinite sum does not converge absolutely, one says the random variable does not have finite expectation. Now consider a random variable X which has a probability density function given by a function f on the real number line . This means that the probability of X taking on

12998-552: The inversion formula: f X ( x ) = 1 2 π ∫ R e − i t x φ X ( t ) d t . {\displaystyle f_{X}(x)={\frac {1}{2\pi }}\int _{\mathbb {R} }e^{-itx}\varphi _{X}(t)\,dt.} For the expected value of g ( X ) {\displaystyle g(X)} (where g : R → R {\displaystyle g:{\mathbb {R} }\to {\mathbb {R} }}

13132-446: The investment security—where holders of securities can sell them to other investors for cash. Otherwise, few people would purchase primary issues, and, thus, companies and governments would be restricted in raising equity capital (money) for their operations. Organized exchanges constitute the main secondary markets. Many smaller issues and most debt securities trade in the decentralized, dealer-based over-the-counter markets. In Europe,

13266-416: The issuer at a discount to resell it at a markup, it is called a firm commitment underwriting . However, if the investment bank considers the risk too great for an underwriting, it may only assent to a best effort agreement , where the investment bank will simply do its best to sell the new issue. For the primary market to thrive, there must be a secondary market , or aftermarket that provides liquidity for

13400-447: The letters "a.s." stand for " almost surely "—a central property of the Lebesgue integral. Basically, one says that an inequality like X ≥ 0 {\displaystyle X\geq 0} is true almost surely, when the probability measure attributes zero-mass to the complementary event { X < 0 } . {\displaystyle \left\{X<0\right\}.} Concentration inequalities control

13534-447: The likelihood of a random variable taking on large values. Markov's inequality is among the best-known and simplest to prove: for a nonnegative random variable X and any positive number a , it states that P ⁡ ( X ≥ a ) ≤ E ⁡ [ X ] a . {\displaystyle \operatorname {P} (X\geq a)\leq {\frac {\operatorname {E} [X]}{a}}.} If X

13668-405: The mantle of "justice." The defense bar generally contends that lower pleading standards allow more frivolous lawsuits and extorted settlements that primarily benefit plaintiff lawyers, not shareholders. On the other hand, the plaintiffs' bar claims that higher pleading standards enable corporate executives to loot their companies and to defraud innocent shareholders with impunity. The importance of

13802-432: The newly abstract situation, this definition is extremely similar in nature to the very simplest definition of expected values, given above, as certain weighted averages. This is because, in measure theory, the value of the Lebesgue integral of X is defined via weighted averages of approximations of X which take on finitely many values. Moreover, if given a random variable with finitely or countably many possible values,

13936-457: The other hand, draw loan funds from the financial resources of the lending institution, not from the sale of the securities. Collateral and sources of collateral are changing, in 2012 gold became a more acceptable form of collateral. By 2015, recently Exchange-traded funds (ETFs) previously seen by many as unpromising had started to become more readily available and acceptable. Public securities markets are either primary or secondary markets. In

14070-423: The pendency of any motion to dismiss." The stay operates against all parties, including both plaintiffs and defendants. The Securities Litigation Uniform Standards Act has amended that provision to allow a federal court to impose the discovery stay on related securities suits in state court. Section 301 adds additional duties to a firm's auditor, including a requirement to report evidence of illegal activity to

14204-407: The plaintiff merely pointing to a false statement and declares that the defendant "must have known" that the statement was false, based upon the position within the company. The United States Supreme Court , in an opinion written by Justice Ruth Bader Ginsburg and issued June 21, 2007, interpreted the standard. The Court ruled that for plaintiffs to allege a "strong inference" of intentional fraud,

14338-502: The possible outcomes of the random variable X and p 1 , p 2 , ... are their corresponding probabilities. In many non-mathematical textbooks, this is presented as the full definition of expected values in this context. However, there are some subtleties with infinite summation, so the above formula is not suitable as a mathematical definition. In particular, the Riemann series theorem of mathematical analysis illustrates that

14472-434: The potential compensatory damages are $ 500 million. Therefore, as a practical matter, if a Rule 10b-5 case is not dismissed on a motion to dismiss and is certified as a class action, the defendant faces a lot of pressure to settle the case. Even if the plaintiff has a relatively weak case, the expected value of going to trial will put the defendant under pressure to settle. That is, the cost of high-enough-damages, even when it

14606-415: The price of the stock in question has declined substantially. By then, the "bad facts" about the company have already been absorbed by the market and so plaintiffs cannot show that the disclosure of the analyst's dishonesty caused any further decline in the stock price. To prevent discovery until a court has deemed a securities fraud complaint sufficient, the PSLRA also imposes a stay of all discovery "during

14740-536: The primary market, the money for the securities is received by the issuer of the securities from investors, typically in an initial public offering (IPO). In the secondary market, the securities are simply assets held by one investor selling them to another investor, with the money going from one investor to the other. An initial public offering is when a company issues public stock newly to investors, called an "IPO" for short. A company can later issue more new shares, or issue shares that have been previously registered in

14874-455: The prospect of capital growth. Equity investment may also offer control of the business of the issuer. Debt holdings may also offer some measure of control to the investor if the company is a fledgling start-up or an old giant undergoing restructuring . In these cases, if interest payments are missed, the creditors may take control of the company and liquidate it to recover some of their investment. The last decade has seen an enormous growth in

15008-416: The public investing personally, other than by way of business. In distinction, the greatest part of investment in terms of volume, is wholesale , i.e., by financial institutions acting on their own account, or on behalf of clients. Important institutional investors include investment banks , insurance companies, pension funds and other managed funds. The "wholesaler" is typically an underwriter or

15142-672: The random variables. To see this, let U {\displaystyle U} be a random variable distributed uniformly on [ 0 , 1 ] . {\displaystyle [0,1].} For n ≥ 1 , {\displaystyle n\geq 1,} define a sequence of random variables X n = n ⋅ 1 { U ∈ ( 0 , 1 n ) } , {\displaystyle X_{n}=n\cdot \mathbf {1} \left\{U\in \left(0,{\tfrac {1}{n}}\right)\right\},} with 1 { A } {\displaystyle \mathbf {1} \{A\}} being

15276-537: The ratification of the PSLRA in 1995 can be fully appreciated only when by noting such long-running debate between the plaintiff bar and the defense bar over the applicable standards governing Rule 12(b)(6) motions to dismiss in Rule 10b-5 claims. The legislative history reveals that Congress passed the PSLRA to curb the "abusive practices committed in private securities litigation" including "the routine filing of lawsuits against issuers of securities and others whenever there

15410-415: The reluctance of judges to impose Section 11 sanctions as an additional reason for the passage of the legislation. The PSLRA does impose some new rules on securities class action lawsuits. It allows judges to decide the most adequate plaintiff in class actions by favoring institutional investors with large dollar amounts at stake. It mandates full disclosure to investors of proposed settlements, including

15544-465: The report made to the Board. That provision constitutes an important affirmative requirement of auditors to breach confidentiality. Although PSLRA reduced the public protections from fraudulent operations within a public company that issue from fraudster's aversion to civil liability, the burden of decreasing fraud risk is shifted by PSLRA to auditors of public firms, which presumably have deeper access to critical financial operations and details. The PSLRA

15678-402: The rules that govern such motion to dismiss. Plaintiff lawyers advocate less demanding standards, as such standards would result in a larger number of cases to proceed to discovery and higher amounts of settlements. Defense lawyers and the corporations that they represent, on the other hand, advocate more demanding standards, for opposite reasons. Both sides in the debate use rhetoric that claims

15812-438: The same finite area, i.e. if ∫ − ∞ μ F ( x ) d x = ∫ μ ∞ ( 1 − F ( x ) ) d x {\displaystyle \int _{-\infty }^{\mu }F(x)\,dx=\int _{\mu }^{\infty }{\big (}1-F(x){\big )}\,dx} and both improper Riemann integrals converge. Finally, this

15946-449: The same fundamental principle. The principle is that the value of a future gain should be directly proportional to the chance of getting it. This principle seemed to have come naturally to both of them. They were very pleased by the fact that they had found essentially the same solution, and this in turn made them absolutely convinced that they had solved the problem conclusively; however, they did not publish their findings. They only informed

16080-418: The same principle. But finally I have found that my answers in many cases do not differ from theirs. In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematically in terms of the expectations of random variables . Neither Pascal nor Huygens used the term "expectation" in its modern sense. In particular, Huygens writes: That any one Chance or Expectation to win any thing

16214-540: The same rights and privileges as shares held in certificated form. Bearer securities are completely negotiable and entitle the holder to the rights under the security (e.g., to payment if it is a debt security, and voting if it is an equity security). They are transferred by delivering the instrument from person to person. In some cases, transfer is by endorsement, or signing the back of the instrument, and delivery. Regulatory and fiscal authorities sometimes regard bearer securities negatively, as they may be used to facilitate

16348-518: The strong inference requirement. And, the Seventh Circuit... did not consider competing inferences, but concluded that if an inference of culpability exists, the pleading is sufficient. Finally, the PSLRA also makes clear that a plaintiff in a Rule 10b-5 case "shall have the burden of proving that the act or omission of the defendant... caused the loss for which the plaintiff seeks to recover damages" (15 U.S.C. § 78u-4(b)(4)). At one time, there

16482-514: The sum hoped for. We will call this advantage mathematical hope. The use of the letter E to denote "expected value" goes back to W. A. Whitworth in 1901. The symbol has since become popular for English writers. In German, E stands for Erwartungswert , in Spanish for esperanza matemática , and in French for espérance mathématique. When "E" is used to denote "expected value", authors use

16616-441: The term "security" to refer to any form of financial instrument, even though the underlying legal and regulatory regime may not have such a broad definition. In some jurisdictions the term specifically excludes financial instruments other than equity and fixed income instruments. In some jurisdictions it includes some instruments that are close to equities and fixed income, e.g., equity warrants . Securities may be represented by

16750-464: The term is used differently by various authors. Analogously to the countably-infinite case above, there are subtleties with this expression due to the infinite region of integration. Such subtleties can be seen concretely if the distribution of X is given by the Cauchy distribution Cauchy(0, π) , so that f ( x ) = ( x + π ) . It is straightforward to compute in this case that ∫

16884-411: The theory of chance is the product of the sum hoped for by the probability of obtaining it; it is the partial sum which ought to result when we do not wish to run the risks of the event in supposing that the division is made proportional to the probabilities. This division is the only equitable one when all strange circumstances are eliminated; because an equal degree of probability gives an equal right for

17018-544: The traditional business of stock exchanges. Large volumes of securities are also bought and sold "over the counter" (OTC). OTC dealing involves buyers and sellers dealing with each other by telephone or electronically on the basis of prices that are displayed electronically, usually by financial data vendors such as SuperDerivatives, Reuters , Investing.com and Bloomberg . There are also eurosecurities, which are securities that are issued outside their domestic market into more than one jurisdiction. They are generally listed on

17152-408: The traditional method used by commercial enterprises to raise new capital. They may offer an attractive alternative to bank loans - depending on their pricing and market demand for particular characteristics. A disadvantage of bank loans as a source of financing is that the bank may seek a measure of protection against default by the borrower via extensive financial covenants. Through securities, capital

17286-1010: The use of securities as collateral . Purchasing securities with borrowed money secured by other securities or cash itself is called " buying on margin ". Where A is owed a debt or other obligation by B, A may require B to deliver property rights in securities to A, either at inception (transfer of title) or only in default (non-transfer-of-title institutional). For institutional loans, property rights are not transferred but nevertheless enable A to satisfy its claims in case B fails to make good on its obligations to A or otherwise becomes insolvent . Collateral arrangements are divided into two broad categories, namely security interests and outright collateral transfers. Commonly, commercial banks, investment banks, government agencies and other institutional investors such as mutual funds are significant collateral takers as well as providers. In addition, private parties may utilize stocks or other securities as collateral for portfolio loans in securities lending scenarios. On

17420-411: The value of certain infinite sums involving positive and negative summands depends on the order in which the summands are given. Since the outcomes of a random variable have no naturally given order, this creates a difficulty in defining expected value precisely. For this reason, many mathematical textbooks only consider the case that the infinite sum given above converges absolutely , which implies that

17554-548: Was a question over whether the requirement actually applied at the motion to dismiss stage. The statute says that the plaintiff must "prove" loss causation, but it does not say the plaintiff must "allege" loss causation. However, after the Supreme Court's decision in Dura Pharmaceuticals v. Broudo , it is now clear that a plaintiff must allege loss causation in its complaint. The Dura decision held that

17688-421: Was enacted into law by the U.S. Congress over a veto by President Bill Clinton . The U.S. House of Representatives approved the bill by a 319–100 margin, and the U.S. Senate approved it, 68–30. Every Republican in the House voted in favor of the legislation, and only four Republicans in the Senate voted against it: William Cohen , John McCain , Richard Shelby , and Arlen Specter . Prominent liberals in

17822-414: Was false (the legal term of art for that state of mind is scienter). In alleging scienter under the PSLRA, the plaintiff must, "with respect to each act or omission alleged to violate this chapter, state with particularity facts giving rise to a strong inference that the defendant acted with the required state of mind." 15 U.S.C. § 78u-4(b)(2). The requirement allows defendants to obtain dismissal of cases of

17956-434: Was when it came to its application to the real world. Pascal, being a mathematician, was provoked and determined to solve the problem once and for all. He began to discuss the problem in the famous series of letters to Pierre de Fermat . Soon enough, they both independently came up with a solution. They solved the problem in different computational ways, but their results were identical because their computations were based on

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