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82-532: Statistics commonly deals with random samples. A random sample can be thought of as a set of objects that are chosen randomly. More formally, it is "a sequence of independent, identically distributed (IID) random data points." In other words, the terms random sample and IID are synonymous. In statistics, " random sample " is the typical terminology, but in probability, it is more common to say " IID ." Independent and identically distributed random variables are often used as an assumption, which tends to simplify

164-543: A n d   B ) = P ( A ) P ( B ) {\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}P({\color {red}A}\ \mathrm {and} \ {\color {green}B})=P({\color {red}A})P({\color {green}B})} . In the following, P ( A B ) {\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}P({\color {red}A}{\color {green}B})}

246-402: A Bayesian network or copula functions . When two or more random variables are defined on a probability space, it is useful to describe how they vary together; that is, it is useful to measure the relationship between the variables. A common measure of the relationship between two random variables is the covariance. Covariance is a measure of linear relationship between the random variables. If

328-401: A linear regression model, like this: height i  = b 0  + b 1 age i  + ε i , where b 0 is the intercept, b 1 is a parameter that age is multiplied by to obtain a prediction of height, ε i is the error term, and i identifies the child. This implies that height is predicted by age, with some error. An admissible model must be consistent with all

410-437: A die 10 times and record how many times the result is 1. Choose a card from a standard deck of cards containing 52 cards, then place the card back in the deck. Repeat this 52 times. Record the number of kings that appear. Many results that were first proven under the assumption that the random variables are i.i.d . have been shown to be true even under a weaker distributional assumption. The most general notion which shares

492-423: A fair or unfair roulette wheel is i.i.d . One implication of this is that if the roulette ball lands on "red", for example, 20 times in a row, the next spin is no more or less likely to be "black" than on any other spin (see the gambler's fallacy ). Toss a coin 10 times and record how many times the coin lands on heads. Such a sequence of two possible i.i.d. outcomes is also called a Bernoulli process . Roll

574-402: A linear function of age; that errors in the approximation are distributed as i.i.d. Gaussian. The assumptions are sufficient to specify P {\displaystyle {\mathcal {P}}} —as they are required to do. A statistical model is a special class of mathematical model . What distinguishes a statistical model from other mathematical models is that a statistical model

656-413: A margin of the table. Consider the flip of two fair coins ; let A {\displaystyle A} and B {\displaystyle B} be discrete random variables associated with the outcomes of the first and second coin flips respectively. Each coin flip is a Bernoulli trial and has a Bernoulli distribution . If a coin displays "heads" then the associated random variable takes

738-450: A mixture of arbitrary numbers of discrete and continuous random variables. In general two random variables X {\displaystyle X} and Y {\displaystyle Y} are independent if and only if the joint cumulative distribution function satisfies Two discrete random variables X {\displaystyle X} and Y {\displaystyle Y} are independent if and only if

820-518: A shorter notation: The joint probability mass function of two discrete random variables X , Y {\displaystyle X,Y} is: or written in terms of conditional distributions where P ( Y = y ∣ X = x ) {\displaystyle \mathrm {P} (Y=y\mid X=x)} is the probability of Y = y {\displaystyle Y=y} given that X = x {\displaystyle X=x} . The generalization of

902-428: A situation in which one may wish to find the cumulative distribution of one random variable which is continuous and another random variable which is discrete arises when one wishes to use a logistic regression in predicting the probability of a binary outcome Y conditional on the value of a continuously distributed outcome X {\displaystyle X} . One must use the "mixed" joint density when finding

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984-451: A statistical model ( S , P {\displaystyle S,{\mathcal {P}}} ) with P = { F θ : θ ∈ Θ } {\displaystyle {\mathcal {P}}=\{F_{\theta }:\theta \in \Theta \}} . In notation, we write that Θ ⊆ R k {\displaystyle \Theta \subseteq \mathbb {R} ^{k}} where k

1066-440: A statistical model is "a formal representation of a theory" ( Herman Adèr quoting Kenneth Bollen ). Informally, a statistical model can be thought of as a statistical assumption (or set of statistical assumptions) with a certain property: that the assumption allows us to calculate the probability of any event . As an example, consider a pair of ordinary six-sided dice . We will study two different statistical assumptions about

1148-480: A statistical model is a pair ( S , P {\displaystyle S,{\mathcal {P}}} ), where S {\displaystyle S} is the set of possible observations, i.e. the sample space , and P {\displaystyle {\mathcal {P}}} is a set of probability distributions on S {\displaystyle S} . The set P {\displaystyle {\mathcal {P}}} represents all of

1230-568: A useful generalization — for example, sampling without replacement is not independent, but is exchangeable. In stochastic calculus , i.i.d. variables are thought of as a discrete time Lévy process : each variable gives how much one changes from one time to another. For example, a sequence of Bernoulli trials is interpreted as the Bernoulli process . One may generalize this to include continuous time Lévy processes , and many Lévy processes can be seen as limits of i.i.d. variables—for instance,

1312-657: Is semiparametric if it has both finite-dimensional and infinite-dimensional parameters. Formally, if k is the dimension of Θ {\displaystyle \Theta } and n is the number of samples, both semiparametric and nonparametric models have k → ∞ {\displaystyle k\rightarrow \infty } as n → ∞ {\displaystyle n\rightarrow \infty } . If k / n → 0 {\displaystyle k/n\rightarrow 0} as n → ∞ {\displaystyle n\rightarrow \infty } , then

1394-475: Is conditionally dependent given another subset B {\displaystyle B} of these variables, then the probability mass function of the joint distribution is P ( X 1 , … , X n ) {\displaystyle \mathrm {P} (X_{1},\ldots ,X_{n})} . P ( X 1 , … , X n ) {\displaystyle \mathrm {P} (X_{1},\ldots ,X_{n})}

1476-420: Is 1. If more than one random variable is defined in a random experiment, it is important to distinguish between the joint probability distribution of X and Y and the probability distribution of each variable individually. The individual probability distribution of a random variable is referred to as its marginal probability distribution. In general, the marginal probability distribution of X can be determined from

1558-458: Is a positive integer ( R {\displaystyle \mathbb {R} } denotes the real numbers ; other sets can be used, in principle). Here, k is called the dimension of the model. The model is said to be parametric if Θ {\displaystyle \Theta } has finite dimension. As an example, if we assume that data arise from a univariate Gaussian distribution , then we are assuming that In this example,

1640-430: Is a possibility P ( B | A ) {\textstyle P({\color {green}B}|{\color {red}A})} . Generally, the occurrence of A {\textstyle \color {red}A} has an effect on the probability of B {\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\color {Green}B} — this

1722-1243: Is called conditional probability. Additionally, only when the occurrence of A {\textstyle \color {red}A} has no effect on the occurrence of B {\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\color {Green}B} , there is P ( B | A ) = P ( B ) {\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}P({\color {green}B}|{\color {red}A})=P({\color {green}B})} . Note: If P ( A ) > 0 {\textstyle P({\color {red}A})>0} and P ( B ) > 0 {\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}P({\color {Green}B})>0} , then A {\textstyle \color {red}A} and B {\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\color {Green}B} are mutually independent which cannot be established with mutually incompatible at

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1804-834: Is defined as the derivative of the joint cumulative distribution function (see Eq.1 ): This is equal to: where f Y ∣ X ( y ∣ x ) {\displaystyle f_{Y\mid X}(y\mid x)} and f X ∣ Y ( x ∣ y ) {\displaystyle f_{X\mid Y}(x\mid y)} are the conditional distributions of Y {\displaystyle Y} given X = x {\displaystyle X=x} and of X {\displaystyle X} given Y = y {\displaystyle Y=y} respectively, and f X ( x ) {\displaystyle f_{X}(x)} and f Y ( y ) {\displaystyle f_{Y}(y)} are

1886-460: Is equal to P ( B ) ⋅ P ( A ∣ B ) {\displaystyle P(B)\cdot P(A\mid B)} . Therefore, it can be efficiently represented by the lower-dimensional probability distributions P ( B ) {\displaystyle P(B)} and P ( A ∣ B ) {\displaystyle P(A\mid B)} . Such conditional independence relations can be represented with

1968-420: Is fundamental for much of statistical inference . Konishi & Kitagawa (2008 , p. 75) state: "The majority of the problems in statistical inference can be considered to be problems related to statistical modeling. They are typically formulated as comparisons of several statistical models." Common criteria for comparing models include the following: R , Bayes factor , Akaike information criterion , and

2050-416: Is non- deterministic . Thus, in a statistical model specified via mathematical equations, some of the variables do not have specific values, but instead have probability distributions; i.e. some of the variables are stochastic . In the above example with children's heights, ε is a stochastic variable; without that stochastic variable, the model would be deterministic. Statistical models are often used even when

2132-427: Is often easier to interpret than the covariance. The correlation just scales the covariance by the product of the standard deviation of each variable. Consequently, the correlation is a dimensionless quantity that can be used to compare the linear relationships between pairs of variables in different units. If the points in the joint probability distribution of X and Y that receive positive probability tend to fall along

2214-665: Is short for P ( A   a n d   B ) {\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}P({\color {red}A}\ \mathrm {and} \ {\color {green}B})} . Suppose there are two events of the experiment, A {\textstyle \color {red}A} and B {\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\color {Green}B} . If P ( A ) > 0 {\textstyle P({\color {red}A})>0} , there

2296-414: Is the probability density function of ( X , Y ) {\displaystyle (X,Y)} with respect to the product measure on the respective supports of X {\displaystyle X} and Y {\displaystyle Y} . Either of these two decompositions can then be used to recover the joint cumulative distribution function: The definition generalizes to

2378-638: Is the set of all possible values of θ {\displaystyle \theta } , then P = { F θ : θ ∈ Θ } {\displaystyle {\mathcal {P}}=\{F_{\theta }:\theta \in \Theta \}} . (The parameterization is identifiable, and this is easy to check.) In this example, the model is determined by (1) specifying S {\displaystyle S} and (2) making some assumptions relevant to P {\displaystyle {\mathcal {P}}} . There are two assumptions: that height can be approximated by

2460-576: The Wiener process is the limit of the Bernoulli process. Machine learning (ML) involves learning statistical relationships within data. To train ML models effectively, it is crucial to use data that is broadly generalizable. If the training data is insufficiently representative of the task, the model's performance on new, unseen data may be poor. The i.i.d. hypothesis allows for a significant reduction in

2542-2486: The cumulative distribution functions of X {\displaystyle X} and Y {\displaystyle Y} , respectively, and denote their joint cumulative distribution function by F X , Y ( x , y ) = P ⁡ ( X ≤ x ∧ Y ≤ y ) {\displaystyle F_{X,Y}(x,y)=\operatorname {P} (X\leq x\land Y\leq y)} . Two random variables X {\displaystyle X} and Y {\displaystyle Y} are identically distributed if and only if F X ( x ) = F Y ( x ) ∀ x ∈ I {\displaystyle F_{X}(x)=F_{Y}(x)\,\forall x\in I} . Two random variables X {\displaystyle X} and Y {\displaystyle Y} are independent if and only if F X , Y ( x , y ) = F X ( x ) ⋅ F Y ( y ) ∀ x , y ∈ I {\displaystyle F_{X,Y}(x,y)=F_{X}(x)\cdot F_{Y}(y)\,\forall x,y\in I} . (See further Independence (probability theory) § Two random variables .) Two random variables X {\displaystyle X} and Y {\displaystyle Y} are i.i.d. if they are independent and identically distributed, i.e. if and only if The definition extends naturally to more than two random variables. We say that n {\displaystyle n} random variables X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} are i.i.d. if they are independent (see further Independence (probability theory) § More than two random variables ) and identically distributed, i.e. if and only if where F X 1 , … , X n ( x 1 , … , x n ) = P ⁡ ( X 1 ≤ x 1 ∧ … ∧ X n ≤ x n ) {\displaystyle F_{X_{1},\ldots ,X_{n}}(x_{1},\ldots ,x_{n})=\operatorname {P} (X_{1}\leq x_{1}\land \ldots \land X_{n}\leq x_{n})} denotes

Independent and identically distributed random variables - Misplaced Pages Continue

2624-475: The data-generating process . When referring specifically to probabilities , the corresponding term is probabilistic model . All statistical hypothesis tests and all statistical estimators are derived via statistical models. More generally, statistical models are part of the foundation of statistical inference . A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables. As such,

2706-579: The likelihood-ratio test together with its generalization, the relative likelihood . Another way of comparing two statistical models is through the notion of deficiency introduced by Lucien Le Cam . Joint probability distribution Given two random variables that are defined on the same probability space , the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered for any given number of random variables. The joint distribution encodes

2788-473: The marginal distributions for X {\displaystyle X} and Y {\displaystyle Y} respectively. The definition extends naturally to more than two random variables: Again, since these are probability distributions, one has respectively The "mixed joint density" may be defined where one or more random variables are continuous and the other random variables are discrete. With one variable of each type One example of

2870-408: The marginal distributions , i.e. the distributions of each of the individual random variables and the conditional probability distributions , which deal with how the outputs of one random variable are distributed when given information on the outputs of the other random variable(s). In the formal mathematical setup of measure theory , the joint distribution is given by the pushforward measure , by

2952-504: The probability that the random variable X {\displaystyle X} takes on a value less than or equal to x {\displaystyle x} and that Y {\displaystyle Y} takes on a value less than or equal to y {\displaystyle y} . For N {\displaystyle N} random variables X 1 , … , X N {\displaystyle X_{1},\ldots ,X_{N}} ,

3034-464: The context of sequences of random variables. Then, "independent and identically distributed" implies that an element in the sequence is independent of the random variables that came before it. In this way, an i.i.d. sequence is different from a Markov sequence , where the probability distribution for the n th random variable is a function of the previous random variable in the sequence (for a first-order Markov sequence). An i.i.d. sequence does not imply

3116-405: The cumulative distribution of this binary outcome because the input variables ( X , Y ) {\displaystyle (X,Y)} were initially defined in such a way that one could not collectively assign it either a probability density function or a probability mass function. Formally, f X , Y ( x , y ) {\displaystyle f_{X,Y}(x,y)}

3198-453: The data points. Thus, a straight line (height i  = b 0  + b 1 age i ) cannot be admissible for a model of the data—unless it exactly fits all the data points, i.e. all the data points lie perfectly on the line. The error term, ε i , must be included in the equation, so that the model is consistent with all the data points. To do statistical inference , we would first need to assume some probability distributions for

3280-414: The data-generating process being modeled is deterministic. For instance, coin tossing is, in principle, a deterministic process; yet it is commonly modeled as stochastic (via a Bernoulli process ). Choosing an appropriate statistical model to represent a given data-generating process is sometimes extremely difficult, and may require knowledge of both the process and relevant statistical analyses. Relatedly,

3362-404: The dice. The first statistical assumption is this: for each of the dice, the probability of each face (1, 2, 3, 4, 5, and 6) coming up is ⁠ 1 / 6 ⁠ . From that assumption, we can calculate the probability of both dice coming up 5:  ⁠ 1 / 6 ⁠ × ⁠ 1 / 6 ⁠  =   ⁠ 1 / 36 ⁠ .  More generally, we can calculate

Independent and identically distributed random variables - Misplaced Pages Continue

3444-403: The dimension, k , equals 2. As another example, suppose that the data consists of points ( x , y ) that we assume are distributed according to a straight line with i.i.d. Gaussian residuals (with zero mean): this leads to the same statistical model as was used in the example with children's heights. The dimension of the statistical model is 3: the intercept of the line, the slope of the line, and

3526-657: The events A {\textstyle \color {red}A} , B {\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\color {Green}B} , and C {\textstyle \definecolor {blue}{rgb}{0,0,1}\color {blue}C} are mutually independent. A more general definition is there are n {\textstyle n} events, A 1 , A 2 , … , A n {\textstyle {\color {red}A}_{1},{\color {red}A}_{2},\ldots ,{\color {red}A}_{n}} . If

3608-433: The example above, with the first assumption, calculating the probability of an event is easy. With some other examples, though, the calculation can be difficult, or even impractical (e.g. it might require millions of years of computation). For an assumption to constitute a statistical model, such difficulty is acceptable: doing the calculation does not need to be practicable, just theoretically possible. In mathematical terms,

3690-423: The first integral is over all points in the range of (X,Y) for which X=x and the second integral is over all points in the range of (X,Y) for which Y=y. For a pair of random variables X , Y {\displaystyle X,Y} , the joint cumulative distribution function (CDF) F X , Y {\displaystyle F_{X,Y}} is given by where the right-hand side represents

3772-405: The first model can be transformed into the second model by imposing constraints on the parameters of the first model. As an example, the set of all Gaussian distributions has, nested within it, the set of zero-mean Gaussian distributions: we constrain the mean in the set of all Gaussian distributions to get the zero-mean distributions. As a second example, the quadratic model has, nested within it,

3854-444: The joint CDF F X 1 , … , X N {\displaystyle F_{X_{1},\ldots ,X_{N}}} is given by Interpreting the N {\displaystyle N} random variables as a random vector X = ( X 1 , … , X N ) T {\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{N})^{T}} yields

3936-503: The joint cumulative distribution function of X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} . In probability theory, two events, A {\textstyle \color {red}A} and B {\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\color {Green}B} , are called independent if and only if P ( A  

4018-431: The joint distribution. In any one cell the probability of a particular combination occurring is (since the draws are independent) the product of the probability of the specified result for A and the probability of the specified result for B. The probabilities in these four cells sum to 1, as with all probability distributions. Moreover, the final row and the final column give the marginal probability distribution for A and

4100-742: The joint probability distribution of X and other random variables. If the joint probability density function of random variable X and Y is f X , Y ( x , y ) {\displaystyle f_{X,Y}(x,y)} , the marginal probability density function of X and Y, which defines the marginal distribution , is given by: f X ( x ) = ∫ f X , Y ( x , y ) d y {\displaystyle f_{X}(x)=\int f_{X,Y}(x,y)\;dy} f Y ( y ) = ∫ f X , Y ( x , y ) d x {\displaystyle f_{Y}(y)=\int f_{X,Y}(x,y)\;dx} where

4182-445: The joint probability mass function becomes Since the coin flips are independent, the joint probability mass function is the product of the marginals: Consider the roll of a fair die and let A = 1 {\displaystyle A=1} if the number is even (i.e. 2, 4, or 6) and A = 0 {\displaystyle A=0} otherwise. Furthermore, let B = 1 {\displaystyle B=1} if

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4264-553: The joint probability mass function satisfies for all x {\displaystyle x} and y {\displaystyle y} . While the number of independent random events grows, the related joint probability value decreases rapidly to zero, according to a negative exponential law. Similarly, two absolutely continuous random variables are independent if and only if for all x {\displaystyle x} and y {\displaystyle y} . This means that acquiring any information about

4346-480: The linear model —we constrain the parameter b 2 to equal 0. In both those examples, the first model has a higher dimension than the second model (for the first example, the zero-mean model has dimension 1). Such is often, but not always, the case. As an example where they have the same dimension, the set of positive-mean Gaussian distributions is nested within the set of all Gaussian distributions; they both have dimension 2. Comparing statistical models

4428-401: The main properties of i.i.d. variables are exchangeable random variables , introduced by Bruno de Finetti . Exchangeability means that while variables may not be independent, future ones behave like past ones — formally, any value of a finite sequence is as likely as any permutation of those values — the joint probability distribution is invariant under the symmetric group . This provides

4510-409: The map obtained by pairing together the given random variables, of the sample space's probability measure . In the case of real-valued random variables, the joint distribution, as a particular multivariate distribution, may be expressed by a multivariate cumulative distribution function , or by a multivariate probability density function together with a multivariate probability mass function . In

4592-449: The mapping is injective ), it is said to be identifiable . In some cases, the model can be more complex. Suppose that we have a population of children, with the ages of the children distributed uniformly , in the population. The height of a child will be stochastically related to the age: e.g. when we know that a child is of age 7, this influences the chance of the child being 1.5 meters tall. We could formalize that relationship in

4674-466: The marginal probability distribution for B respectively. For example, for A the first of these cells gives the sum of the probabilities for A being red, regardless of which possibility for B in the column above the cell occurs, as 2/3. Thus the marginal probability distribution for A {\displaystyle A} gives A {\displaystyle A} 's probabilities unconditional on B {\displaystyle B} , in

4756-402: The model is semiparametric; otherwise, the model is nonparametric. Parametric models are by far the most commonly used statistical models. Regarding semiparametric and nonparametric models, Sir David Cox has said, "These typically involve fewer assumptions of structure and distributional form but usually contain strong assumptions about independencies". Two statistical models are nested if

4838-751: The models that are considered possible. This set is typically parameterized: P = { F θ : θ ∈ Θ } {\displaystyle {\mathcal {P}}=\{F_{\theta }:\theta \in \Theta \}} . The set Θ {\displaystyle \Theta } defines the parameters of the model. If a parameterization is such that distinct parameter values give rise to distinct distributions, i.e. F θ 1 = F θ 2 ⇒ θ 1 = θ 2 {\displaystyle F_{\theta _{1}}=F_{\theta _{2}}\Rightarrow \theta _{1}=\theta _{2}} (in other words,

4920-483: The number is prime (i.e. 2, 3, or 5) and B = 0 {\displaystyle B=0} otherwise. Then, the joint distribution of A {\displaystyle A} and B {\displaystyle B} , expressed as a probability mass function, is These probabilities necessarily sum to 1, since the probability of some combination of A {\displaystyle A} and B {\displaystyle B} occurring

5002-899: The number of individual cases required in the training sample, simplifying optimization calculations. In optimization problems, the assumption of independent and identical distribution simplifies the calculation of the likelihood function. Due to this assumption, the likelihood function can be expressed as: l ( θ ) = P ( x 1 , x 2 , x 3 , . . . , x n | θ ) = P ( x 1 | θ ) P ( x 2 | θ ) P ( x 3 | θ ) . . . P ( x n | θ ) {\displaystyle l(\theta )=P(x_{1},x_{2},x_{3},...,x_{n}|\theta )=P(x_{1}|\theta )P(x_{2}|\theta )P(x_{3}|\theta )...P(x_{n}|\theta )} To maximize

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5084-401: The outcomes of the draw from the first urn and second urn respectively. The probability of drawing a red ball from either of the urns is 2/3, and the probability of drawing a blue ball is 1/3. The joint probability distribution is presented in the following table: Each of the four inner cells shows the probability of a particular combination of results from the two draws; these probabilities are

5166-419: The preceding two-variable case is the joint probability distribution of n {\displaystyle n\,} discrete random variables X 1 , X 2 , … , X n {\displaystyle X_{1},X_{2},\dots ,X_{n}} which is: or equivalently This identity is known as the chain rule of probability . Since these are probabilities, in

5248-407: The probabilities for all elements of the sample space or event space must be the same. For example, repeated throws of loaded dice will produce a sequence that is i.i.d., despite the outcomes being biased. In signal processing and image processing , the notion of transformation to i.i.d. implies two specifications, the "i.d." part and the "i." part: i.d . – The signal level must be balanced on

5330-502: The probabilities of the product events for any 2 , 3 , … , n {\textstyle 2,3,\ldots ,n} events are equal to the product of the probabilities of each event, then the events A 1 , A 2 , … , A n {\textstyle {\color {red}A}_{1},{\color {red}A}_{2},\ldots ,{\color {red}A}_{n}} are independent of each other. A sequence of outcomes of spins of

5412-480: The probability of any event: e.g. (1 and 2) or (3 and 3) or (5 and 6). The alternative statistical assumption is this: for each of the dice, the probability of the face 5 coming up is ⁠ 1 / 8 ⁠ (because the dice are weighted ). From that assumption, we can calculate the probability of both dice coming up 5:  ⁠ 1 / 8 ⁠ × ⁠ 1 / 8 ⁠  =   ⁠ 1 / 64 ⁠ .  We cannot, however, calculate

5494-421: The probability of any other nontrivial event, as the probabilities of the other faces are unknown. The first statistical assumption constitutes a statistical model: because with the assumption alone, we can calculate the probability of any event. The alternative statistical assumption does not constitute a statistical model: because with the assumption alone, we cannot calculate the probability of every event. In

5576-1205: The probability of the observed event, the log function is applied to maximize the parameter θ {\textstyle \theta } . Specifically, it computes: a r g m a x θ ⁡ log ⁡ ( l ( θ ) ) {\displaystyle \mathop {\rm {argmax}} \limits _{\theta }\log(l(\theta ))} where log ⁡ ( l ( θ ) ) = log ⁡ ( P ( x 1 | θ ) ) + log ⁡ ( P ( x 2 | θ ) ) + log ⁡ ( P ( x 3 | θ ) ) + . . . + log ⁡ ( P ( x n | θ ) ) {\displaystyle \log(l(\theta ))=\log(P(x_{1}|\theta ))+\log(P(x_{2}|\theta ))+\log(P(x_{3}|\theta ))+...+\log(P(x_{n}|\theta ))} Computers are very efficient at performing multiple additions, but not as efficient at performing multiplications. This simplification enhances computational efficiency. The log transformation, in

5658-487: The process of maximizing, converts many exponential functions into linear functions. There are two main reasons why this hypothesis is practically useful with the central limit theorem (CLT): Statistical modeling A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population ). A statistical model represents, often in considerably idealized form,

5740-416: The relationship between the random variables is nonlinear, the covariance might not be sensitive to the relationship, which means, it does not relate the correlation between two variables. The covariance between the random variables X {\displaystyle X} and Y {\displaystyle Y} is There is another measure of the relationship between two random variables that

5822-1843: The same time; that is, independence must be compatible and mutual exclusion must be related. Suppose A {\textstyle \color {red}A} , B {\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\color {Green}B} , and C {\textstyle \definecolor {blue}{rgb}{0,0,1}\color {blue}C} are three events. If P ( A B ) = P ( A ) P ( B ) {\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}P({\color {red}A}{\color {green}B})=P({\color {red}A})P({\color {green}B})} , P ( B C ) = P ( B ) P ( C ) {\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\definecolor {blue}{rgb}{0,0,1}\definecolor {Blue}{rgb}{0,0,1}P({\color {green}B}{\color {blue}C})=P({\color {green}B})P({\color {blue}C})} , P ( A C ) = P ( A ) P ( C ) {\textstyle \definecolor {blue}{rgb}{0,0,1}P({\color {red}A}{\color {blue}C})=P({\color {red}A})P({\color {blue}C})} , and P ( A B C ) = P ( A ) P ( B ) P ( C ) {\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\definecolor {blue}{rgb}{0,0,1}\definecolor {Blue}{rgb}{0,0,1}P({\color {red}A}{\color {green}B}{\color {blue}C})=P({\color {red}A})P({\color {green}B})P({\color {blue}C})} are satisfied, then

5904-405: The set of all possible pairs (age, height). Each possible value of θ {\displaystyle \theta }  = ( b 0 , b 1 , σ ) determines a distribution on S {\displaystyle S} ; denote that distribution by F θ {\displaystyle F_{\theta }} . If Θ {\displaystyle \Theta }

5986-537: The special case of continuous random variables , it is sufficient to consider probability density functions, and in the case of discrete random variables , it is sufficient to consider probability mass functions. Each of two urns contains twice as many red balls as blue balls, and no others, and one ball is randomly selected from each urn, with the two draws independent of each other. Let A {\displaystyle A} and B {\displaystyle B} be discrete random variables associated with

6068-438: The statistician Sir David Cox has said, "How [the] translation from subject-matter problem to statistical model is done is often the most critical part of an analysis". There are three purposes for a statistical model, according to Konishi & Kitagawa: Those three purposes are essentially the same as the three purposes indicated by Friendly & Meyer: prediction, estimation, description. Suppose that we have

6150-799: The time axis. i . – The signal spectrum must be flattened, i.e. transformed by filtering (such as deconvolution ) to a white noise signal (i.e. a signal where all frequencies are equally present). Suppose that the random variables X {\displaystyle X} and Y {\displaystyle Y} are defined to assume values in I ⊆ R {\displaystyle I\subseteq \mathbb {R} } . Let F X ( x ) = P ⁡ ( X ≤ x ) {\displaystyle F_{X}(x)=\operatorname {P} (X\leq x)} and F Y ( y ) = P ⁡ ( Y ≤ y ) {\displaystyle F_{Y}(y)=\operatorname {P} (Y\leq y)} be

6232-457: The two-variable case which generalizes for n {\displaystyle n\,} discrete random variables X 1 , X 2 , … , X n {\displaystyle X_{1},X_{2},\dots ,X_{n}} to The joint probability density function f X , Y ( x , y ) {\displaystyle f_{X,Y}(x,y)} for two continuous random variables

6314-402: The underlying mathematics. In practical applications of statistical modeling , however, this assumption may or may not be realistic. The i.i.d. assumption is also used in the central limit theorem , which states that the probability distribution of the sum (or average) of i.i.d. variables with finite variance approaches a normal distribution . The i.i.d. assumption frequently arises in

6396-412: The univariate Gaussian distribution, θ {\displaystyle \theta } is formally a single parameter with dimension 2, but it is often regarded as comprising 2 separate parameters—the mean and the standard deviation. A statistical model is nonparametric if the parameter set Θ {\displaystyle \Theta } is infinite dimensional. A statistical model

6478-403: The value 1, and it takes the value 0 otherwise. The probability of each of these outcomes is 1/2, so the marginal (unconditional) density functions are The joint probability mass function of A {\displaystyle A} and B {\displaystyle B} defines probabilities for each pair of outcomes. All possible outcomes are Since each outcome is equally likely

6560-425: The value of one or more of the random variables leads to a conditional distribution of any other variable that is identical to its unconditional (marginal) distribution; thus no variable provides any information about any other variable. If a subset A {\displaystyle A} of the variables X 1 , ⋯ , X n {\displaystyle X_{1},\cdots ,X_{n}}

6642-400: The variance of the distribution of the residuals. (Note the set of all possible lines has dimension 2, even though geometrically, a line has dimension 1.) Although formally θ ∈ Θ {\displaystyle \theta \in \Theta } is a single parameter that has dimension k , it is sometimes regarded as comprising k separate parameters. For example, with

6724-520: The ε i . For instance, we might assume that the ε i distributions are i.i.d. Gaussian, with zero mean. In this instance, the model would have 3 parameters: b 0 , b 1 , and the variance of the Gaussian distribution. We can formally specify the model in the form ( S , P {\displaystyle S,{\mathcal {P}}} ) as follows. The sample space, S {\displaystyle S} , of our model comprises

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