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1Chapter 12 - Joint Probability Density Function and Cumulative Distribution Function The joint probability density function (PDF) is a function fx,y(x,y) such that for any set C of pairs of real numbers (x,y) the PDF is given by? The joint cumulative distribution function (CDF) is?Pr((X,Y) element of C)=double integral[x,y element of c] fx,y(x,y) dxdy
Fx,y(x,y) = Pr(X
2Chapter 12 - Marginal PDFs and Independent Random Variables The marginal PDF of X is? The marginal PDF of Y is? X and Y are independent if?fx(x) = integral [from -infinity to infinity] fx,y(x,y)dy
fy(y) = integral [from -infinity to infinity] fx,y(x,y)dx
fx,y(x,y) = fx(x)fy(y) or
Fx,y(x,y) = Fx(x)Fy(y)
3Chapter 12 - Conditional Distributions The conditional distribution of X given Y=y is? The conditional distribution of Y given X=x is? X and Y are independent if?fx¦y(x¦y) = fx,y(x,y)/fy(y)
fy¦x(y¦x) = fx,y(x,y)/fx(x)
fx¦y(x¦y) = fx(x) or fy¦x(y¦x) = fy(y)