Transcription of 4.2 Conditional Distributions and Independence
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4.2 Conditional Distributions and Independence
Conditional Distributions and IndependenceDefinition (X, Y) be a discrete bivariate random vector with joint pmff(x, y) andmarginal pmfsfX(x) andfY(y). For anyxsuch thatP(X=x) =fX(x)>0, the conditionalpmf ofYgiven thatX=xis the function ofydenoted byf(y|x) and defined byf(y|x) =P(Y=y|X=x) =f(x, y)fX(x).For anyysuch thatP(Y=y) =fY(y)>0, the Conditional pmf ofXgiven thatY=yisthe function ofxdenoted byf(x|y) and defined byf(x|y) =P(X=x|Y=y) =f(x, y)fY(y).It is easy to verify thatf(y|x) andf(x|y) are indeed Distributions .
4.2 Conditional Distributions and Independence Definition 4.2.1 Let (X,Y) be a discrete bivariate random vector with joint pmf f(x,y) andmarginal pmfs fX(x) and fY (y).For any x such that P(X = x) = fX(x) > 0, the conditional pmf of Y given that X = x is the function of y denoted by f(y|x) and defined by f(y|x) = P(Y = y|X = x) = f(x,y) fX(x) For any y such that P(Y = y) = fY (y) > 0, the ...
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