Random variable

A random variable is a rigorously defined mathematical entity used mainly to describe chance and probability in a mathematical way. Probability is the likelihood or chance that something is the case or will happen The structure of random variables was developed and formalized to simplify the analysis of games of chance, stochastic events, and the results of scientific experiments by retaining only the mathematical properties necessary to answer probabilistic questions. A game of chance is a Game whose outcome is strongly influenced by some randomizing device and upon which contestants frequently wager money Stochastic (from the Greek "Στόχος" for "aim" or "guess" means Random. In scientific inquiry an experiment ( Latin: Ex- periri, "to try out" is a method of investigating particular types of research questions or Further formalizations have firmly grounded the entity in the theoretical domains of mathematics by making use of measure theory. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with

Fortunately, the language and structure of random variables can be grasped at various levels of mathematical fluency. Set theory and calculus are fundamental. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives

Broadly, there are two types of random variable — discrete and continuous. Discrete random variables take on one of a set of specific values, each with some probability greater than zero. Continuous random variables can be realized with any of a range of values (e. g. , a real number between zero and one), and so there are several ranges (e. g. 0 to one half) that have a probability greater than zero of occurring.

A random variable has either an associated probability distribution (discrete random variable) or probability density function (continuous random variable).

1 Intuitive definition
- 1.1 Examples
2 Measure theory definition
3 Real-valued random variables
- 3.1 Distribution functions of random variables
- 3.2 Moments
4 Functions of random variables
- 4.1 Example 1
- 4.2 Example 2
5 Equivalence of random variables
6 Convergence
7 Literature
8 See also

Intuitive definition

Intuitively, a random variable is thought of as a function mapping the sample space of a random process to the real numbers. A few examples will highlight this.

Examples

For a coin toss, the possible events are heads or tails. The number of heads appearing in one fair coin toss can be described using the following random variable:

$X = \begin{cases}1,& \text{if heads} ,\\ 0,& \text{if tails} .\end{cases}$

with probability mass function given by:

$\rho_X(x) = \begin{cases}\frac{1}{2},& \text{if x=0} ,\\ \frac{1}{2},& \text{if x=1},\\ 0,& \text{otherwise} .\end{cases}$

A random variable can also be used to describe the process of rolling a fair die and the possible outcomes. In Probability theory, a probability mass function (abbreviated pmf) is a function that gives the probability that a discrete Random variable For other uses see either Die or Dice (disambiguation. Dice (the Plural of Die, from Old French The most obvious representation is to take the set { 1, 2, 3, 4, 5, 6 } as the sample space, defining the random variable X as the number rolled. In Probability theory, the sample space or universal sample space, often denoted S, Ω or U (for "universe" of an Experiment In this case,

$X = \begin{cases}1,& \text{if a 1 is rolled} ,\\ 2,& \text{if a 2 is rolled} ,\\ 3,& \text{if a 3 is rolled} ,\\ 4,& \text{if a 4 is rolled} ,\\ 5,& \text{if a 5 is rolled} ,\\ 6,& \text{if a 6 is rolled} .\end{cases}$

$\rho_X(x) = \begin{cases}\frac{1}{6},& \text{if x=1,2,3,4,5,6} ,\\ 0,& \text{otherwise} .\end{cases}$

Measure theory definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space and (Y, Σ) be a measurable space. A probability space, in Probability theory, is the conventional Mathematical model of Randomness. In Mathematics, a σ-algebra (or sigma-algebra) ( Sigma is a Greek letter upper case Σ lower case σ over a set X is a nonempty Then a random variable X is formally defined as a measurable function $X: \Omega \rightarrow Y$ . In Mathematics, measurable functions are Well-behaved functions between measurable spaces. An interpretation of this is that the preimage of the "well-behaved" subsets of Y (the elements of Σ) are events (elements of $\mathcal{F}$ ), and hence are assigned a probability by P. In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage

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

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

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. The Heaviside step function, H, also called the unit step function, is a discontinuous function whose value is zero for negative In general, the CDF specifies the probability that the variable takes on particular values.

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 $\{ s \in\Omega : X(s) > 2 \}$ 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. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable 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

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

and sometimes also using a probability density function. In Probability theory and Statistics, the cumulative distribution function (CDF, also probability distribution function or just distribution function In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability In measure-theoretic terms, we use the random variable X to "push-forward" the measure P on Ω to a measure dF on R. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with 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.

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 and Statistics, the variance of a Random variable, Probability distribution, or sample is one measure of In Probability and Statistics, the standard deviation is a measure of the dispersion of a collection of 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 &mu to the sequences of moments

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. In Mathematics, measurable functions are Well-behaved functions between measurable spaces. 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

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

Example 1

Let X be a real-valued, continuous random variable and let Y = X². In Probability theory and Statistics, the cumulative distribution function (CDF, also probability distribution function or just distribution function In Probability theory, a Probability distribution 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

$F_Y(y) = 0\qquad\hbox{if}\quad y < 0.$

If y ≥ 0, then

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

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

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:

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

Two random variables having equal moment generating functions have the same distribution. In Probability theory and Statistics, the moment-generating function of a Random variable X is M_X(t=\operatorname{E}\left(e^{tX}\right This provides, for example, a useful method of checking equality of certain functions of i.i.d. random variables. "IID" or "iid" redirects here For other uses see IID (disambiguation.

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

which is the basis of the Kolmogorov-Smirnov test. In Statistics, the Kolmogorov &ndash Smirnov test (also called the K-S test for brevity is a form of Minimum distance estimation used

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,

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

As in the previous case, there is a related distance between the random variables, namely

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

This is equivalent to the following:

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. In Mathematics, the concepts of essential supremum and essential infimum are related to the notions of Supremum and Infimum, but the former are In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with

Equality

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

$X(\omega)=Y(\omega)\qquad\hbox{for all}\quad\omega$

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 an ordered list of objects (or events The law of large numbers (LLN is a theorem in Probability that describes the long-term stability of the mean of a Random variable. The central limit theorem (CLT states that the sum of a sufficiently large number of identically distributed independent Random variables each with finite

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 The convergence (in one of the senses presented below of Sequences

Literature

Kallenberg, O., Random Measures, 4th edition. Olav Kallenberg is a Physicist and Mathematician living in Auburn AL, USA 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.

Dictionary

-noun

(statistics) (broadly) A quantity whose values are random and to which a probability distribution is assigned, such as the possible outcomes of a roll of a dice.
(statistics) (formally) A measurable function from a sample space to the measurable space of possible values of the variable.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents