Random variable

A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. For example, 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 }. Another random variable might describe the possible outcomes of picking a random person and measuring their height.

operatorname{P}(-sqrt{y} le X le sqrt{y}),

so

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

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Moments

The probability distribution of 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. Note that in general, E is not the same as f(E). 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.

Related Topics:
Expected value - Variance - Standard deviation

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Mathematically, this is known as the (generalised) problem of moments: for a given class of random variables X, find a collection {fi} of functions such that the expectation values E fully characterize the distribution of the random variable X.

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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.

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

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Equality in distribution

Two random variables X and Y are equal in distribution if

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

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To be equal in distribution, random variables need not be defined on the same probability space, but without loss of generality they can be made into independent random variables on the same probability space. The notion of equivalence in distribution is associated to the following notion of distance between probability distributions,

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

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which is the basis of the Kolmogorov-Smirnov test.

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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,

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

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Equality in pth mean implies equality in qth mean for all q

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

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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:

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:operatorname{P}(X eq Y) = 0.

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

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

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where 'sup' in this case represents the essential supremum in the sense of measure theory.

Related Topics:
Essential supremum - Measure theory

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Equality

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

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

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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.

Related Topics:
Sequence - Law of large numbers - Central limit theorem

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There are various senses in which a sequence (Xn) of random variables can converge to a random variable X. These are explained in the article on convergence of random variables.

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See also

discrete random variable, continuous random variable, probability distribution, randomness, random vector, random function, generating function. Algorithmic information theory

Related Topics:
Discrete random variable - Continuous random variable - Probability distribution - Randomness - Random vector - Random function - Generating function - Algorithmic information theory

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