Problem 4. It is known that A and B are independent events that satisfy lP'(A) = % and lP'(A0 n B) = %. Calculate 1P(B). Show solution Problem 5. Assume that X is a random variable whose moment generating function satisfies the following: For every t e (00, 10) 100 Assume that Y is a random variable independent from X such that for every real number t the following holds: 1P(X S t) = IP'(Y S t). Evaluate 1E[(2X + 3Y) - (3X 7Y)]. Show solution Problem 6. Eight boys and seven girls went to movies and sat in the same row of 15 seats. Assuming that all the 15! permutations of their seating arrangements are equally probable, compute the expected number of pairs of neighbors of different genders