Abalancedcoinisflippedtwice.Let X denote theoutcomeofthefirstflipand Y denote the outcomeofthesecondflip,representing head by1and tail by0.Supposetheflipsare independent.

(a) Let Z indicate whetherbothflipshadthesameresult,with z = 1 for yes and z = 0 for no.

Showthat X and Z are independent.

(b) Showthatalthough X and Y are independent, X and Z are independent,and Y and Z are independent, X, Y , and Z are notmutuallyindependent.So,ifasetofrandom variablesare pairwiseindependent, theyarenotnecessarily mutuallyindependent.