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Unformatted text preview: 1 Probability Density Functions (PDF) For a continuous RV X with PDF f X ( x ) ( 0), P ( a X b ) = integraldisplay b a f X ( x ) dx P ( x X x + ) f X ( x ) . P ( X A ) = integraldisplay A f X ( x ) dx Remarks: if X is continuous, P ( X = x ) = 0 x !! f X ( x ) may take values larger than 1. Normalization property: integraldisplay  f X ( x ) dx = 1 2 2 Mean and variance of a continuous RV E [ X ] = integraldisplay  xf X ( x ) dx E [ g ( X )] = integraldisplay  g ( x ) f X ( x ) dx Var( X ) = integraldisplay  ( x E [ X ]) 2 f X ( x ) dx = E [ X 2 ] ( E [ X ]) 2 ( 0) E [ aX + b ] = a E [ X ] + b Var( aX + b ) = a 2 Var( X ) 3 3 Cumulative Distribution Functions Definition: F X ( x ) = P ( X x ) monotonically increasing from 0 (at ) to 1 (at + ). Continuous RV: F X ( x ) = P ( X x ) = integraldisplay x f X ( t ) dt (continuous) f X ( x ) = dF X dx ( x ) Discrete RV: F X ( x ) = P ( X x ) = summationdisplay k x p X ( k ) (piecewise constant) p X ( k ) = F X ( k ) F X ( k 1) 4 4 Normal/Gaussian Random Variables Standard Normal RV: N (0 , 1): f X ( x ) = 1 2 e x 2 / 2 E [ X ] = 0 , Var( X ) = 1 General normal RV: N ( , 2 ): f X ( x ) = 1 2 e ( x ) 2 / 2 2 E [ X ] = , Var( X ) = 2 5 if Y = aX + b , then Y N ( a + b, a 2 2 ). CDF for standard normal ( . ) can be read in a table. To evaluate CDF of a general standard normal, express it as a function of a standard normal: X N ( , 2 ) X N (0 , 1) P ( X x ) = P parenleftBig X x parenrightBig = parenleftBig x parenrightBig where ( . ) denotes the CDF of a standard normal. 6 5 Joint PDF Joint PDF of two continuous RV X and Y : f X,Y ( x,y ). P ( x X x + , y Y y + ) f X,Y ( x,y ) . 2 P ( A ) = integraldisplay integraldisplay A f X,Y ( x,y ) dxdy E [ g ( X,Y )] = integraldisplay  integraldisplay  g ( x,y...
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 Spring '07
 BARD

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