Misplaced Pages

#592407

69-604: MLN may refer to: Markov logic network Midlothian , historic county in Scotland, Chapman code Minuteman Library Network Modern Language Notes , a US journal of European literature Melilla Airport , Melilla, Spain, IATA code Movement of National Liberation , Mexico, 1960s Former National Liberation Movement (Guatemala) National Liberation Movement (Panama) See also [ edit ] National Liberation Movement (disambiguation) Topics referred to by

138-410: A b f ( x ) d x . {\displaystyle P\left(a\leq X\leq b\right)=\int _{a}^{b}f(x)\,dx.} This is the definition of a probability density function , so that absolutely continuous probability distributions are exactly those with a probability density function. In particular, the probability for X {\displaystyle X} to take any single value

207-401: A {\displaystyle a} (that is, a ≤ X ≤ a {\displaystyle a\leq X\leq a} ) is zero, because an integral with coinciding upper and lower limits is always equal to zero. If the interval [ a , b ] {\displaystyle [a,b]} is replaced by any measurable set A {\displaystyle A} ,

276-439: A , b ] ⊂ R {\displaystyle I=[a,b]\subset \mathbb {R} } the probability of X {\displaystyle X} belonging to I {\displaystyle I} is given by the integral of f {\displaystyle f} over I {\displaystyle I} : P ( a ≤ X ≤ b ) = ∫

345-460: A measurable space ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} . Given that probabilities of events of the form { ω ∈ Ω ∣ X ( ω ) ∈ A } {\displaystyle \{\omega \in \Omega \mid X(\omega )\in A\}} satisfy Kolmogorov's probability axioms ,

414-400: A probability distribution is the mathematical function that gives the probabilities of occurrence of possible outcomes for an experiment . It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events ( subsets of the sample space). For instance, if X is used to denote the outcome of a coin toss ("the experiment"), then

483-495: A Bernoulli distribution with parameter p {\displaystyle p} . This is a transformation of discrete random variable. For a distribution function F {\displaystyle F} of an absolutely continuous random variable, an absolutely continuous random variable must be constructed. F i n v {\displaystyle F^{\mathit {inv}}} , an inverse function of F {\displaystyle F} , relates to

552-414: A coin toss, a roll of a die) and the probabilities are encoded by a discrete list of the probabilities of the outcomes; in this case the discrete probability distribution is known as probability mass function . On the other hand, absolutely continuous probability distributions are applicable to scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as

621-1109: A discrete probability distribution, there is a countable set A {\displaystyle A} with P ( X ∈ A ) = 1 {\displaystyle P(X\in A)=1} and a probability mass function p {\displaystyle p} . If E {\displaystyle E} is any event, then P ( X ∈ E ) = ∑ ω ∈ A p ( ω ) δ ω ( E ) , {\displaystyle P(X\in E)=\sum _{\omega \in A}p(\omega )\delta _{\omega }(E),} or in short, P X = ∑ ω ∈ A p ( ω ) δ ω . {\displaystyle P_{X}=\sum _{\omega \in A}p(\omega )\delta _{\omega }.} Similarly, discrete distributions can be represented with

690-1024: A discrete random variable X {\displaystyle X} , let u 0 , u 1 , … {\displaystyle u_{0},u_{1},\dots } be the values it can take with non-zero probability. Denote Ω i = X − 1 ( u i ) = { ω : X ( ω ) = u i } , i = 0 , 1 , 2 , … {\displaystyle \Omega _{i}=X^{-1}(u_{i})=\{\omega :X(\omega )=u_{i}\},\,i=0,1,2,\dots } These are disjoint sets , and for such sets P ( ⋃ i Ω i ) = ∑ i P ( Ω i ) = ∑ i P ( X = u i ) = 1. {\displaystyle P\left(\bigcup _{i}\Omega _{i}\right)=\sum _{i}P(\Omega _{i})=\sum _{i}P(X=u_{i})=1.} It follows that

759-468: A given domain, a Markov logic network defines a probability distribution on the set of all interpretations of its predicates on the given domain. The underlying idea is that an interpretation is more likely if it satisfies formulas with positive weights and less likely if it satisfies formulas with negative weights. For any n {\displaystyle n} -ary predicate symbol R {\displaystyle R} that occurs in

SECTION 10

#1732776277593

828-467: A more general definition of density functions and the equivalent absolutely continuous measures see absolutely continuous measure . In the measure-theoretic formalization of probability theory , a random variable is defined as a measurable function X {\displaystyle X} from a probability space ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )} to

897-418: A multivariate distribution (a joint probability distribution ) gives the probabilities of a random vector – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution , the hypergeometric distribution , and the normal distribution . A commonly encountered multivariate distribution

966-534: A myriad of phenomena, since most practical distributions are supported on relatively simple subsets, such as hypercubes or balls . However, this is not always the case, and there exist phenomena with supports that are actually complicated curves γ : [ a , b ] → R n {\displaystyle \gamma :[a,b]\rightarrow \mathbb {R} ^{n}} within some space R n {\displaystyle \mathbb {R} ^{n}} or similar. In these cases,

1035-433: A number in [ 0 , 1 ] ⊆ R {\displaystyle [0,1]\subseteq \mathbb {R} } . The probability function P {\displaystyle P} can take as argument subsets of the sample space itself, as in the coin toss example, where the function P {\displaystyle P} was defined so that P (heads) = 0.5 and P (tails) = 0.5 . However, because of

1104-1172: A popular formalism for statistical relational learning . A Markov logic network consists of a collection of formulas from first-order logic , to each of which is assigned a real number , the weight. The underlying idea is that an interpretation is more likely if it satisfies formulas with positive weights and less likely if it satisfies formulas with negative weights. For instance, the following Markov logic network codifies how smokers are more likely to be friends with other smokers, and how stress encourages smoking: 2.0 :: s m o k e s ( X ) ← s m o k e s ( Y ) ∧ i n f l u e n c e s ( X , Y ) 0.5 :: s m o k e s ( X ) ← s t r e s s ( X ) {\displaystyle {\begin{array}{lcl}2.0&::&\mathrm {smokes} (X)\leftarrow \mathrm {smokes} (Y)\land \mathrm {influences} (X,Y)\\0.5&::&\mathrm {smokes} (X)\leftarrow \mathrm {stress} (X)\end{array}}} Together with

1173-421: A random variable takes values from a continuum then by convention, any individual outcome is assigned probability zero. For such continuous random variables , only events that include infinitely many outcomes such as intervals have probability greater than 0. For example, consider measuring the weight of a piece of ham in the supermarket, and assume the scale can provide arbitrarily many digits of precision. Then,

1242-414: A set of vectors , a set of arbitrary non-numerical values, etc. For example, the sample space of a coin flip could be  Ω = { "heads", "tails" } . To define probability distributions for the specific case of random variables (so the sample space can be seen as a numeric set), it is common to distinguish between discrete and absolutely continuous random variables . In the discrete case, it

1311-425: A set of probability zero, where 1 A {\displaystyle 1_{A}} is the indicator function of A {\displaystyle A} . This may serve as an alternative definition of discrete random variables. A special case is the discrete distribution of a random variable that can take on only one fixed value; in other words, it is a deterministic distribution . Expressed formally,

1380-400: A sine, sin ⁡ ( t ) {\displaystyle \sin(t)} , whose limit when t → ∞ {\displaystyle t\rightarrow \infty } does not converge. Formally, the measure exists only if the limit of the relative frequency converges when the system is observed into the infinite future. The branch of dynamical systems that studies

1449-778: A uniform distribution between 0 and 1. To construct a random Bernoulli variable for some 0 < p < 1 {\displaystyle 0<p<1} , we define X = { 1 , if  U < p 0 , if  U ≥ p {\displaystyle X={\begin{cases}1,&{\text{if }}U<p\\0,&{\text{if }}U\geq p\end{cases}}} so that Pr ( X = 1 ) = Pr ( U < p ) = p , Pr ( X = 0 ) = Pr ( U ≥ p ) = 1 − p . {\displaystyle \Pr(X=1)=\Pr(U<p)=p,\quad \Pr(X=0)=\Pr(U\geq p)=1-p.} This random variable X has

SECTION 20

#1732776277593

1518-403: Is a countable set with P ( X ∈ A ) = 1 {\displaystyle P(X\in A)=1} . Thus the discrete random variables (i.e. random variables whose probability distribution is discrete) are exactly those with a probability mass function p ( x ) = P ( X = x ) {\displaystyle p(x)=P(X=x)} . In the case where

1587-399: Is a mathematical description of the probabilities of events, subsets of the sample space . The sample space, often represented in notation by   Ω   , {\displaystyle \ \Omega \ ,} is the set of all possible outcomes of a random phenomenon being observed. The sample space may be any set: a set of real numbers , a set of descriptive labels,

1656-553: Is a probability distribution on the real numbers with uncountably many possible values, such as a whole interval in the real line, and where the probability of any event can be expressed as an integral. More precisely, a real random variable X {\displaystyle X} has an absolutely continuous probability distribution if there is a function f : R → [ 0 , ∞ ] {\displaystyle f:\mathbb {R} \to [0,\infty ]} such that for each interval I = [

1725-560: Is common to denote as P ( X ∈ E ) {\displaystyle P(X\in E)} the probability that a certain value of the variable X {\displaystyle X} belongs to a certain event E {\displaystyle E} . The above probability function only characterizes a probability distribution if it satisfies all the Kolmogorov axioms , that is: The concept of probability function

1794-401: Is defined as F ( x ) = P ( X ≤ x ) . {\displaystyle F(x)=P(X\leq x).} The cumulative distribution function of any real-valued random variable has the properties: Conversely, any function F : R → R {\displaystyle F:\mathbb {R} \to \mathbb {R} } that satisfies the first four of

1863-438: Is different from Wikidata All article disambiguation pages All disambiguation pages Markov logic network In 2002, Ben Taskar , Pieter Abbeel and Daphne Koller introduced relational Markov networks as templates to specify Markov networks abstractly and without reference to a specific domain . Work on Markov logic networks began in 2003 by Pedro Domingos and Matt Richardson. Markov logic networks

1932-612: Is given by allocating a Boolean truth value ( true or false ) to each grounding of an element. A true grounding of a formula φ {\displaystyle \varphi } in an interpretation with free variables x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} is a variable assignment of x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} that makes φ {\displaystyle \varphi } true in that interpretation. Then

2001-508: Is made more rigorous by defining it as the element of a probability space ( X , A , P ) {\displaystyle (X,{\mathcal {A}},P)} , where X {\displaystyle X} is the set of possible outcomes, A {\displaystyle {\mathcal {A}}} is the set of all subsets E ⊂ X {\displaystyle E\subset X} whose probability can be measured, and P {\displaystyle P}

2070-462: Is possible because this measurement does not require as much precision from the underlying equipment. Absolutely continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval. An alternative description of

2139-420: Is studied, the observed states from the subset are as indicated in red. So one could ask what is the probability of observing a state in a certain position of the red subset; if such a probability exists, it is called the probability measure of the system. This kind of complicated support appears quite frequently in dynamical systems . It is not simple to establish that the system has a probability measure, and

MLN - Misplaced Pages Continue

2208-455: Is sufficient to specify a probability mass function   p   {\displaystyle \ p\ } assigning a probability to each possible outcome (e.g. when throwing a fair die , each of the six digits “1” to “6” , corresponding to the number of dots on the die, has the probability   1 6   ) . {\displaystyle \ {\tfrac {1}{6}}~).} The probability of an event

2277-424: Is the multivariate normal distribution . Besides the probability function, the cumulative distribution function, the probability mass function and the probability density function, the moment generating function and the characteristic function also serve to identify a probability distribution, as they uniquely determine an underlying cumulative distribution function. Some key concepts and terms, widely used in

2346-406: Is the area under the probability density function from   − ∞   {\displaystyle \ -\infty \ } to   x   , {\displaystyle \ x\ ,} as shown in figure 1. A probability distribution can be described in various forms, such as by a probability mass function or a cumulative distribution function. One of

2415-516: Is the number of its true groundings. This can also be seen as inducing a Markov network whose nodes are the groundings of the predicates occurring in the Markov logic network. The feature functions of this network are the groundings of the sentences occurring in the Markov logic network, with value e w {\displaystyle e^{w}} if the grounding is true and 1 otherwise (where again w {\displaystyle w}

2484-632: Is the probability distribution of a random variable that can take on only a countable number of values ( almost surely ) which means that the probability of any event E {\displaystyle E} can be expressed as a (finite or countably infinite ) sum: P ( X ∈ E ) = ∑ ω ∈ A ∩ E P ( X = ω ) , {\displaystyle P(X\in E)=\sum _{\omega \in A\cap E}P(X=\omega ),} where A {\displaystyle A}

2553-399: Is the probability function, or probability measure , that assigns a probability to each of these measurable subsets E ∈ A {\displaystyle E\in {\mathcal {A}}} . Probability distributions usually belong to one of two classes. A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g.

2622-421: Is the weight of the formula). The probability distributions induced by Markov logic networks can be queried for the probability of a particular event, given by an atomic formula ( marginal inference ), possibly conditioned by another atomic formula. Marginal inference can be performed using standard Markov network inference techniques over the minimal subset of the relevant Markov network required for answering

2691-666: Is then defined to be the sum of the probabilities of all outcomes that satisfy the event; for example, the probability of the event "the die rolls an even value" is   p ( “ 2 ” ) + p ( “ 4 ” ) + p ( “ 6 ” ) = 1 6 + 1 6 + 1 6 = 1 2   . {\displaystyle \ p({\text{“}}2{\text{”}})+p({\text{“}}4{\text{”}})+p({\text{“}}6{\text{”}})={\tfrac {1}{6}}+{\tfrac {1}{6}}+{\tfrac {1}{6}}={\tfrac {1}{2}}~.} In contrast, when

2760-1175: The Dirac delta function as a generalized probability density function f {\displaystyle f} , where f ( x ) = ∑ ω ∈ A p ( ω ) δ ( x − ω ) , {\displaystyle f(x)=\sum _{\omega \in A}p(\omega )\delta (x-\omega ),} which means P ( X ∈ E ) = ∫ E f ( x ) d x = ∑ ω ∈ A p ( ω ) ∫ E δ ( x − ω ) = ∑ ω ∈ A ∩ E p ( ω ) {\displaystyle P(X\in E)=\int _{E}f(x)\,dx=\sum _{\omega \in A}p(\omega )\int _{E}\delta (x-\omega )=\sum _{\omega \in A\cap E}p(\omega )} for any event E . {\displaystyle E.} For

2829-520: The Poisson distribution , the Bernoulli distribution , the binomial distribution , the geometric distribution , the negative binomial distribution and categorical distribution . When a sample (a set of observations) is drawn from a larger population, the sample points have an empirical distribution that is discrete, and which provides information about the population distribution. Additionally,

MLN - Misplaced Pages Continue

2898-473: The discrete uniform distribution is commonly used in computer programs that make equal-probability random selections between a number of choices. A real-valued discrete random variable can equivalently be defined as a random variable whose cumulative distribution function increases only by jump discontinuities —that is, its cdf increases only where it "jumps" to a higher value, and is constant in intervals without jumps. The points where jumps occur are precisely

2967-400: The half-open interval [0, 1) . These random variates X {\displaystyle X} are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated. For example, suppose U {\displaystyle U} has

3036-820: The probability distribution of X {\displaystyle X} is the image measure X ∗ P {\displaystyle X_{*}\mathbb {P} } of X {\displaystyle X} , which is a probability measure on ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} satisfying X ∗ P = P X − 1 {\displaystyle X_{*}\mathbb {P} =\mathbb {P} X^{-1}} . Absolutely continuous and discrete distributions with support on R k {\displaystyle \mathbb {R} ^{k}} or N k {\displaystyle \mathbb {N} ^{k}} are extremely useful to model

3105-448: The Markov logic network and every n {\displaystyle n} - tuple a 1 , … , a n {\displaystyle a_{1},\dots ,a_{n}} of domain elements, R ( a 1 , … , a n ) {\displaystyle R(a_{1},\dots ,a_{n})} is a grounding of R {\displaystyle R} . An interpretation

3174-681: The according equality still holds: P ( X ∈ A ) = ∫ A f ( x ) d x . {\displaystyle P(X\in A)=\int _{A}f(x)\,dx.} An absolutely continuous random variable is a random variable whose probability distribution is absolutely continuous. There are many examples of absolutely continuous probability distributions: normal , uniform , chi-squared , and others . Absolutely continuous probability distributions as defined above are precisely those with an absolutely continuous cumulative distribution function. In this case,

3243-402: The cumulative distribution function F {\displaystyle F} has the form F ( x ) = P ( X ≤ x ) = ∫ − ∞ x f ( t ) d t {\displaystyle F(x)=P(X\leq x)=\int _{-\infty }^{x}f(t)\,dt} where f {\displaystyle f} is a density of

3312-427: The distribution is by means of the cumulative distribution function , which describes the probability that the random variable is no larger than a given value (i.e.,   P ( X < x )   {\displaystyle \ {\boldsymbol {\mathcal {P}}}(X<x)\ } for some   x   {\displaystyle \ x\ } ). The cumulative distribution function

3381-439: The existence of a probability measure is ergodic theory . Note that even in these cases, the probability distribution, if it exists, might still be termed "absolutely continuous" or "discrete" depending on whether the support is uncountable or countable, respectively. Most algorithms are based on a pseudorandom number generator that produces numbers X {\displaystyle X} that are uniformly distributed in

3450-510: The inverse is not true, there exist singular distributions , which are neither absolutely continuous nor discrete nor a mixture of those, and do not have a density. An example is given by the Cantor distribution . Some authors however use the term "continuous distribution" to denote all distributions whose cumulative distribution function is absolutely continuous , i.e. refer to absolutely continuous distributions as continuous distributions. For

3519-449: The literature on the topic of probability distributions, are listed below. In the special case of a real-valued random variable, the probability distribution can equivalently be represented by a cumulative distribution function instead of a probability measure. The cumulative distribution function of a random variable X {\displaystyle X} with regard to a probability distribution p {\displaystyle p}

SECTION 50

#1732776277593

3588-741: The main problem is the following. Let t 1 ≪ t 2 ≪ t 3 {\displaystyle t_{1}\ll t_{2}\ll t_{3}} be instants in time and O {\displaystyle O} a subset of the support; if the probability measure exists for the system, one would expect the frequency of observing states inside set O {\displaystyle O} would be equal in interval [ t 1 , t 2 ] {\displaystyle [t_{1},t_{2}]} and [ t 2 , t 3 ] {\displaystyle [t_{2},t_{3}]} , which might not happen; for example, it could oscillate similar to

3657-437: The most general descriptions, which applies for absolutely continuous and discrete variables, is by means of a probability function P : A → R {\displaystyle P\colon {\mathcal {A}}\to \mathbb {R} } whose input space A {\displaystyle {\mathcal {A}}} is a σ-algebra , and gives a real number probability as its output, particularly,

3726-493: The probability distribution is supported on the image of such curve, and is likely to be determined empirically, rather than finding a closed formula for it. One example is shown in the figure to the right, which displays the evolution of a system of differential equations (commonly known as the Rabinovich–Fabrikant equations ) that can be used to model the behaviour of Langmuir waves in plasma . When this phenomenon

3795-516: The probability distribution of X would take the value 0.5 (1 in 2 or 1/2) for X = heads , and 0.5 for X = tails (assuming that the coin is fair ). More commonly, probability distributions are used to compare the relative occurrence of many different random values. Probability distributions can be defined in different ways and for discrete or for continuous variables. Distributions with special properties or for especially important applications are given specific names. A probability distribution

3864-466: The probability of any given interpretation is directly proportional to exp ⁡ ( ∑ j w j n j ) {\displaystyle \exp(\sum _{j}w_{j}n_{j})} , where w j {\displaystyle w_{j}} is the weight of the j {\displaystyle j} -th sentence of the Markov logic network and n j {\displaystyle n_{j}}

3933-533: The probability that X {\displaystyle X} takes any value except for u 0 , u 1 , … {\displaystyle u_{0},u_{1},\dots } is zero, and thus one can write X {\displaystyle X} as X ( ω ) = ∑ i u i 1 Ω i ( ω ) {\displaystyle X(\omega )=\sum _{i}u_{i}1_{\Omega _{i}}(\omega )} except on

4002-458: The probability that it weighs exactly 500 g must be zero because no matter how high the level of precision chosen, it cannot be assumed that there are no non-zero decimal digits in the remaining omitted digits ignored by the precision level. However, for the same use case, it is possible to meet quality control requirements such as that a package of "500 g" of ham must weigh between 490 g and 510 g with at least 98% probability. This

4071-454: The properties above is the cumulative distribution function of some probability distribution on the real numbers. Any probability distribution can be decomposed as the mixture of a discrete , an absolutely continuous and a singular continuous distribution , and thus any cumulative distribution function admits a decomposition as the convex sum of the three according cumulative distribution functions. A discrete probability distribution

4140-519: The query. Exact inference is known to be #P -complete in the size of the domain. In practice, the exact probability is often approximated. Techniques for approximate inference include Gibbs sampling , belief propagation , or approximation via pseudolikelihood . The class of Markov logic networks which use only two variables in any formula allows for polynomial time exact inference by reduction to weighted model counting . Probability distribution In probability theory and statistics ,

4209-450: The random variable X {\displaystyle X} has a one-point distribution if it has a possible outcome x {\displaystyle x} such that P ( X = x ) = 1. {\displaystyle P(X{=}x)=1.} All other possible outcomes then have probability 0. Its cumulative distribution function jumps immediately from 0 to 1. An absolutely continuous probability distribution

SECTION 60

#1732776277593

4278-408: The random variable X {\displaystyle X} with regard to the distribution P {\displaystyle P} . Note on terminology: Absolutely continuous distributions ought to be distinguished from continuous distributions , which are those having a continuous cumulative distribution function. Every absolutely continuous distribution is a continuous distribution but

4347-643: The range of values is countably infinite, these values have to decline to zero fast enough for the probabilities to add up to 1. For example, if p ( n ) = 1 2 n {\displaystyle p(n)={\tfrac {1}{2^{n}}}} for n = 1 , 2 , . . . {\displaystyle n=1,2,...} , the sum of probabilities would be 1 / 2 + 1 / 4 + 1 / 8 + ⋯ = 1 {\displaystyle 1/2+1/4+1/8+\dots =1} . Well-known discrete probability distributions used in statistical modeling include

4416-494: The real numbers. A discrete probability distribution is often represented with Dirac measures , the probability distributions of deterministic random variables . For any outcome ω {\displaystyle \omega } , let δ ω {\displaystyle \delta _{\omega }} be the Dirac measure concentrated at ω {\displaystyle \omega } . Given

4485-403: The same term [REDACTED] This disambiguation page lists articles associated with the title MLN . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=MLN&oldid=1075551764 " Category : Disambiguation pages Hidden categories: Short description

4554-448: The temperature on a given day. In the absolutely continuous case, probabilities are described by a probability density function , and the probability distribution is by definition the integral of the probability density function. The normal distribution is a commonly encountered absolutely continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time , may demand

4623-432: The use of more general probability measures . A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate , while a distribution whose sample space is a vector space of dimension 2 or more is called multivariate . A univariate distribution gives the probabilities of a single random variable taking on various different values;

4692-463: The values which the random variable may take. Thus the cumulative distribution function has the form F ( x ) = P ( X ≤ x ) = ∑ ω ≤ x p ( ω ) . {\displaystyle F(x)=P(X\leq x)=\sum _{\omega \leq x}p(\omega ).} The points where the cdf jumps always form a countable set; this may be any countable set and thus may even be dense in

4761-439: The widespread use of random variables , which transform the sample space into a set of numbers (e.g., R {\displaystyle \mathbb {R} } , N {\displaystyle \mathbb {N} } ), it is more common to study probability distributions whose argument are subsets of these particular kinds of sets (number sets), and all probability distributions discussed in this article are of this type. It

#592407