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 . First,f(y|x) 0 foreveryysincef(x, y) 0 andfX(x)>0. Second, yf(y|x) = yf(x, y)fX(x)=fX(x)fX(x)= (Calculating Conditional probabilities)Define the joint pmf of (X, Y) byf(0,10) =f(0,20) =218, f(1,10) =f(1,30) =318, f(1,20) =418, f(2,30) = Conditional probabilityfY|X(10|0) =f(0,10)fX(0)=f(0,10)f(0,10) +f(0,20)= (X, Y) be a continuous bivariate random vector with joint pdff(x, y) and marginal pdfsfX(x) andfY(y).
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) and marginal 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 = …
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