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12. Principle of Inclusion and Exclusion for Probability. Let A , A2, ..., An, n 2 2 be a col- lection of events. Show that P(A UA,) = P(A,) + P(A,) - P(A, nA2). From the above result show that P(A, UA, UA,)= P(A ) + P(A2) + P(A;) - P(A, nA,) - P(A, nA;) - P(A, nA;) + P(A, nA, nA;). Hence, using mathematical induction show that 1-1 1-2 n-1 P JA; ) = EPA;) - E EP(A; nA;)+ [ _ _ P(A; nA, nAN) i=1 izl i=l j=it1 i=l j=itl =j+113. Borel-Cantelli Lemma. Let (2, , P) be a probability space and let Aj , Ay, .. . be sets in F. Show that CO NUASUAK mal k=m and hence prove that 1 if A; nA; = 0, i # j and > P(A ) = co ( num ) 0 if EP(A ) > ( has the following probability mass function: P(X = k) = K! -4, k = 0, 1, 2, .. . Show that E(X) = 1 and Var(X) = 1. For a random variable following a binomial distribution, Binomial(n, p), 0