handout_week2_b

handout_week2_b - Stochastic Signals and Systems Random...

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Stochastic Signals and Systems Random Variables Virginia Tech Fall 2008 Distribution Function: Definition Let X be a random variable. The (cumulative) distribution function (cdf) (also known as probability distribution function ) F X ( x ) is defined by F X ( x ) = P [ X x ] for all values x in -∞ ≤ x ≤ ∞ . That is, the cumulative distribution function F X ( x ) is the probability of the event, “The random variable X takes on a value equal to or less than x in a trial of the random experiment.” Note: We could, for example, use the notation F X ( ω ) to represent this function. Specifically, F X ( ω ) = P [ X ω ] .

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Example In the coin-tossing experiment, the probability of heads equals p and the probability of tails equals q . We define the random variable X such that X ( h ) = 1 and X ( t ) = 0. Solution: x < 0, { X x } = {∅} , so that F X ( x ) = 0. 0 x < 1, { X x } = { t } , so that F X ( x ) = P [ { t } ] = 1 - p . x 1, { X x } = { h , t } = S , so that F X ( x ) = 1. Properties 1 F X (+ ) = 1 and F X ( -∞ ) = 0 Proof: F X (+ ) = P [ X + ] = P [ S ] = 1 F X ( -∞ ) = P [ X = -∞ ] = 0 Note: Thus, 0 F X ( x ) 1 2
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