Bridges RANDOM VARIABLES.docxRANDOM VARIABLESFINITE Probability Mass Function (pmf):fX( x )= p( X =x ) Cumulative Distribution Function (cdf):FX( x )=P( X ≤x ) Mean of X or Expected Value of X :μX=E( x )=∑all xx⋅fX( x) Variance of X :V(X)=∑all x(x −μX)2∙ fX(x) If X is a Bernoulli Random Variable, then its distribution is binomial. The Expected Value of a binomial distribution is:E( x )=n⋅pCONTINUOUS Probability Density Function (pdf):fX( x )=P(a≤X ≤b ) Cumulative Distribution Function (cdf):FX( x )=P( X ≤x ) Mean of X or Expected Value of X :μX=E( x )=∫−∞∞x⋅f ( x )dx Variance of X :V(X)=∫−∞∞(x−μX)2∙ fX(x)dx If X is an Exponential Random Variable, thenfX( x )= [0 if x <0(1/α )⋅e(−x /α )if x≥0 FX( x )=[0 if x <01−e(−x/α )if x≥0 If X is an Uniform Random Variable on [a, b], thenBridges RANDOM VARIABLES.docxfX( x )=[0 if x <a1/( b−a) if a≤x≤b0 if x>bFX( x )=[0 if x <a( x−a)/(b−a) if a≤x≤b1 if x>b If X is a Normal Random Variable, then fX(x)=1σX∙√2 π∙
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