... thatYis also astandard normal random variable,X and Yare uncorrelated, butX and Yare not independent.The last claim is easy to see. IfX and Ywere independent, so would beX2 and Y2, but ... function mappingintoIR, and thus technically a random variable,albeit a degenerate one.A random variable mapsintoIR, and we can look at the preimage under the random variable ofsets inIR. ... An=IPA1IPA2IPAn:1.5.4 Correlation and independenceTheorem 5.8 If two random variablesX and Yare independent, and ifg and hare functions fromIRtoIR,thenIE g X hY...