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handout_week3_a - Stochastic Signals and Systems Random...

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Unformatted text preview: Stochastic Signals and Systems Random Variables Virginia Tech Fall 2008 Uniform Random Variable The pdf of the uniform random variable is given by: f X ( x ) = 1 b- a a ≤ x ≤ b x < a and x > b And the cdf is given by: F X ( x ) = x < a x- a b- a a ≤ x ≤ b 1 x > b Gaussian (Normal) Random Variable The pdf for the Gaussian random variable is given by f X ( x ) = 1 √ 2 πσ exp- ( x- m ) 2 2 σ 2 ! where m and σ are the mean and standard deviation, respectively. The Gaussian pdf is a “bell-shaped” curve centered and symmetric about m and whose “width” increases with σ . Gaussian (Normal) Random Variable The cdf for the Gaussian random variable is given by Definition: Q ( ξ ) = 1 √ 2 π Z ∞ ξ e- t 2 / 2 dt (Q-function) erfc ( ξ ) = 2 √ π Z ∞ ξ e- t 2 dt (Complementary error function) erf ( ξ ) = 1- erfc ( ξ ) (Error function) Exponential Random Variable The exponential random variable X with parameter λ has pdf f X ( x ) = x < λ e- λ x x ≥ and cdf F X ( x ) = x < 1- e- λ x x ≥ • The exponential random variable arises in the modeling of the time between occurrence of events (e.g., the time between customer demands for call connections)....
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