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handout_week2_c - Stochastic Signals and Systems Random...

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Stochastic Signals and Systems Random Variables Virginia Tech Fall 2008 Definition The probability density function of X (pdf) , if it exists, is defined as the derivative of F X ( x ) : f X ( x ) = dF x ( x ) dx . Interpretation of f X ( x ) . P [ x < X x + Δ x ] = F X ( x + Δ x ) - F X ( x ) . If F X ( x ) is continuous in its first derivative then, for sufficiently small Δ x , F X ( x + Δ x ) - F X ( x ) = Z x x x f ( ξ ) d ξ f X ( x x Hence for small Δ x : P [ x < X x + Δ x ] f X ( x x Thus f x ( x ) represents the “density” of probability at the point x in the sense that the probability that X is in a small interval in the vicinity of x is approximately f X ( x x .

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Properties 1 f x ( x ) 0. 2 P [ a < X b ] = F X ( b ) - F X ( a ) = Z b -∞ f X ( ξ ) d ξ - Z a -∞ f X ( ξ ) d ξ = Z b a f X ( ξ ) d ξ 3 F X ( x ) = R x -∞ f X ( ξ ) d ξ 4 R -∞ f X ( ξ ) d ξ = F X ( ) - F X ( -∞ ) = 1 . Example A random variable X has pdf f X ( x ) = c ( 1 - x 4 ) - 1 x 1 0 elsewhere . Find c . Find the cdf of X . Find P [ | X | < 1 / 2 ] .
Conditional cdf We define the conditional (cumulative) distribution function of X

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