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Bridges RANDOM VARIABLES.docx RANDOM VARIABLES FINITE Probability Mass Function (pmf): f X ( x) p ( X x) Cumulative Distribution Function (cdf): FX ( x) P( X x) Mean of X or Expected Value of X : X E ( x) x f X ( x) all x 𝑉(𝑋) = ∑𝑎𝑙𝑙 𝑥(𝑥 − 𝜇𝑋 )2 ∙ 𝑓𝑋 (𝑥) Variance of X : If X is a Bernoulli Random Variable, then its distribution is binomial. The Expected Value of a binomial distribution is: E ( x) n p CONTINUOUS f X ( x ) P ( a X b) Probability Density Function (pdf): Cumulative Distribution Function (cdf): FX ( x) P ( X x) Mean of X or Expected Value of X : X E ( x) x f ( x)dx Variance of X : 𝑉(𝑋) = ∫−∞(𝑥 − 𝜇𝑋 )2 ∙ 𝑓𝑋 (𝑥) 𝑑𝑥 ∞ If X is an Exponential Random Variable, then if x 0 0 f X ( x) ( x / ) if x 0 (1/ ) e 0 FX ( x) ( x / ) 1 e if x 0 if x 0 If X is an Uniform Random Variable on [a, b], then 0 if x a f X ( x) 1 /(b a) if a x b 0 if x b 0 if x a FX ( x) ( x a) /(b a) if a x b 1 if x b If X is a Normal Random Variable, then 𝑓𝑋 (𝑥 ) = 1 𝜎𝑋 ∙√2𝜋 𝑥−𝜇 ∙ 2 −0.5( 𝜎 𝑋 ) 𝑋 𝑒