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Encyclopedia > Random variable

In probability theory, a random variable is a quantity whose values are random and to which a probability distribution is assigned. Formally, a random variable is a measurable function from a sample space to the measurable space of possible values of the variable. The formal definition of random variables places experiments involving real-valued outcomes firmly within the measure-theoretic framework and allows us to construct distribution functions of real-valued random variables. Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... In mathematics, measurable functions are well-behaved functions between measurable spaces. ... In probability theory, the sample space, often denoted S, Î© or U (for universe), of an experiment or random trial is the set of all possible outcomes. ... In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... In mathematics, a measure is a function that assigns a number, e. ...

1 Examples
2 Real-valued random variables
- 2.1 Distribution functions of random variables
- 2.2 Moments
3 Functions of random variables
- 3.1 Example 1
- 3.2 Example 2
4 Equivalence of random variables
5 Convergence
6 Literature
7 See also

Examples

A random variable can be used to describe the process of rolling a fair die and the possible outcomes { 1, 2, 3, 4, 5, 6 }. The most obvious representation is to take this set as the sample space, the probability measure to be uniform measure, and the function to be the identity function. Two standard six-sided pipped dice with rounded corners. ... In probability theory, the sample space, often denoted S, Î© or U (for universe), of an experiment or random trial is the set of all possible outcomes. ... In mathematics, a probability space is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...

For a coin toss, a suitable space of possible outcomes is Ω = { H, T } (for heads and tails). An example random variable on this space is

$X(omega) = begin{cases}0,& omega = texttt{H},1,& omega = texttt{T}.end{cases}$

Real-valued random variables

Typically, the measurable space is the measurable space over the real numbers. In this case, let $(Omega, mathcal{F}, P)$ be a probability space. Then, the function $X: Omega rightarrow mathbb{R}$ is a real-valued random variable if In mathematics, the definition of the probability space is the foundation of probability theory. ...

${ omega : X(omega) le r } in mathcal{F} qquad forall r in mathbb{R}$

Distribution functions of random variables

Associating a cumulative distribution function (CDF) with a random variable is a generalization of assigning a value to a variable. If the cdf is a (right continuous) Heaviside step function then the variable takes on the value at the jump with probability 1. In general, the cdf specifies the probability that the variable takes on particular values. The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...

If a random variable $X: Omega to mathbb{R}$ defined on the probability space $(Ω, A, P)$ is given, we can ask questions like "How likely is it that the value of $X$ is bigger than 2?". This is the same as the probability of the event ${ s inOmega : X(s) > 2 }$ which is often written as $P (X > 2)$ for short.

Recording all these probabilities of output ranges of a real-valued random variable X yields the probability distribution of X. The probability distribution "forgets" about the particular probability space used to define X and only records the probabilities of various values of X. Such a probability distribution can always be captured by its cumulative distribution function In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than...

$F_X(x) = operatorname{P}(X le x)$

and sometimes also using a probability density function. In measure-theoretic terms, we use the random variable X to "push-forward" the measure P on Ω to a measure dF on R. The underlying probability space Ω is a technical device used to guarantee the existence of random variables, and sometimes to construct them. In practice, one often disposes of the space Ω altogether and just puts a measure on R that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables. In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... In mathematics, a measure is a function that assigns a number, e. ...

Moments

The probability distribution of a random variable is often characterised by a small number of parameters, which also have a practical interpretation. For example, it is often enough to know what its "average value" is. This is captured by the mathematical concept of expected value of a random variable, denoted E[X]. In general, E[f(X)] is not equal to f(E[X]). Once the "average value" is known, one could then ask how far from this average value the values of X typically are, a question that is answered by the variance and standard deviation of a random variable. In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ...

Mathematically, this is known as the (generalised) problem of moments: for a given class of random variables X, find a collection {f_i} of functions such that the expectation values E[f_i(X)] fully characterize the distribution of the random variable X. In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure μ to the sequences of moments ∫M dμ where M runs over a set of monomials. ...

Functions of random variables

If we have a random variable X on Ω and a measurable function f: R → R, then Y = f(X) will also be a random variable on Ω, since the composition of measurable functions is also measurable. The same procedure that allowed one to go from a probability space (Ω, P) to (R, dF_X) can be used to obtain the distribution of Y. The cumulative distribution function of Y is In mathematics, measurable functions are well-behaved functions between measurable spaces. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than...

$F_Y(y) = operatorname{P}(f(X) le y).$

Example 1

Let X be a real-valued, continuous random variable and let Y = X². Then, By one convention, a random variable X is called continuous if its cumulative distribution function is continuous. ...

$F_Y(y) = operatorname{P}(X^2 le y).$

If y < 0, then P(X² ≤ y) = 0, so

$F_Y(y) = 0qquadhbox{if}quad y < 0.$

If y ≥ 0, then

$operatorname{P}(X^2 le y) = operatorname{P}(|X| le sqrt{y}) = operatorname{P}(-sqrt{y} le X le sqrt{y}),$

$F_Y(y) = F_X(sqrt{y}) - F_X(-sqrt{y})qquadhbox{if}quad y ge 0.$

Example 2

Suppose $scriptstyle X$ is a random variable with a cumulative distribution

$F_{X}(x) = P(X leq x) = frac{1}{(1 + e^{-x})^{theta}}$

where $scriptstyle theta > 0$ is a fixed parameter. Consider the random variable $scriptstyle Y = mathrm{log}(1 + e^{-X}).$ Then,

$F_{Y}(y) = P(Y leq y) = P(mathrm{log}(1 + e^{-X}) leq y) = P(X > -mathrm{log}(e^{y} - 1)).,$

The last expression can be calculated in terms of the cumulative distribution of $X,$ so

$F_{Y}(y) = 1 - F_{X}(-mathrm{log}(e^{y} - 1)) ,$

$= 1 - frac{1}{(1 + e^{mathrm{log}(e^{y} - 1)})^{theta}}$

$= 1 - frac{1}{(1 + e^{y} - 1)^{theta}}$

$= 1 - e^{-y theta}.,$

Equivalence of random variables

There are several different senses in which random variables can be considered to be equivalent. Two random variables can be equal, equal almost surely, equal in mean, or equal in distribution.

In increasing order of strength, the precise definition of these notions of equivalence is given below.

Equality in distribution

Two random variables X and Y are equal in distribution if they have the same distribution functions:

$operatorname{P}(X le x) = operatorname{P}(Y le x)quadhbox{for all}quad x.$

Two random variables having equal moment generating functions have the same distribution. This provides, for example, a useful method of checking equality of certain functions of iidrv's. In probability theory and statistics, the moment-generating function of a random variable X is The moment-generating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith... ...

To be equal in distribution, random variables need not be defined on the same probability space. The notion of equivalence in distribution is associated to the following notion of distance between probability distributions,

$d(X,Y)=sup_x|operatorname{P}(X le x) - operatorname{P}(Y le x)|,$

which is the basis of the Kolmogorov-Smirnov test. In statistics, the Kolmogorov-Smirnov test (often called the K-S test) is used to determine whether two underlying probability distributions differ, or whether an underlying probability distribution differs from a hypothesized distribution, in either case based on finite samples. ...

Equality in mean

Two random variables X and Y are equal in p-th mean if the pth moment of |X − Y| is zero, that is,

$operatorname{E}(|X-Y|^p) = 0.$

Equality in pth mean implies equality in qth mean for all q<p. As in the previous case, there is a related distance between the random variables, namely

$d_p(X, Y) = operatorname{E}(|X-Y|^p).$

Almost sure equality

Two random variables X and Y are equal almost surely if, and only if, the probability that they are different is zero:

$operatorname{P}(X neq Y) = 0.$

For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance:

$d_infty(X,Y)=sup_omega|X(omega)-Y(omega)|,$

where 'sup' in this case represents the essential supremum in the sense of measure theory. The concepts of essential supremum and essential infimum are related to the notions of supremum and infimum, but the former are more relevant in measure theory, where, often times one is not that interested in a property holding all the time, that is for all elements in a set, but... In mathematics, a measure is a function that assigns a number, e. ...

Equality

Finally, the two random variables X and Y are equal if they are equal as functions on their probability space, that is,

$X(omega)=Y(omega)qquadhbox{for all}quadomega$

Convergence

Much of mathematical statistics consists in proving convergence results for certain sequences of random variables; see for instance the law of large numbers and the central limit theorem. In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... // The law of large numbers (LLN) is any of several theorems in probability. ... A central limit theorem is any of a set of weak-convergence results in probability theory. ...

There are various senses in which a sequence (X_n) of random variables can converge to a random variable X. These are explained in the article on convergence of random variables. In probability theory, there exist several different notions of convergence of random variables. ...

Literature

Kallenberg, O., Random Measures, 4th edition. Academic Press, New York, London; Akademie-Verlag, Berlin (1986). MR0854102 ISBN 0123949602
Papoulis, Athanasios 1965 Probability, Random Variables, and Stochastic Processes. McGraw-Hill Kogakusha, Tokyo, 9th edition, ISBN 0-07-119981-0.

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