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.

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, dFX) can be used to obtain the distribution of Y. The cumulative distribution function of Y is

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

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Example

Let X be a real-valued, continuous random variable and let Y = X2. Then,

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

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If y < 0, then P(X2 ≤ y) = 0, so

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

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If y ≥ 0, then

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:operatorname{P}(X^2 le y) = operatorname{P}(|X| le sqrt{y})

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