This is from a pyp but no answers were given, so i would like to know how to solve them. This is under stochastic processes. As i am weak in this, i will ask questions to clear my doubts when i have seen the answers

Question 5. Expectation (20 marks) Let N be the number of distinct words that Victor Hugo knew, and assume these words are numbered 1 to N. Suppose for simplicity that Victor Hugo wrote only 2 novels, A and B. The novels are reasonably long and they are the same length. Let X j be the number of times that word j appears in novel A, and Y) be the number of times it appears in novel B, for 1 S j S N. Assume that X; and Y3- follows a Poisson distribution with the same parameters. (6 marks) (a) Let the number of occurrences of the word \"pieuvre\" in the two novels be independent Poisson distributed with parameter A. Show that the probability that \"pieuvre\" is used in novel B but not in novel A is A2 A3 A4 ,\\ 6 (A+ET1T+W) Hint: We have (6 marks) (b) Assume that A from part (a) is unknown and is itself taken to be a random variable with probability density function f). Let X be the number of times the word \"pieuvre\" appears in novel A and Y be the corresponding value for novel B. Assume that the conditional distribution of X , Y given A is that they are independent Poisson distributed with parameter A. Show that the probability that \"pieuvre\" is used in novel B but not in novel A is the alternating series 1P(X=l)IP(X=2)+IP'(X=3)]P(X=4)+... Hint: Condition on A and use part (a). (8 marks) (c) Assume that every word's numbers of occurrences in A and B are distributed as in (b), where A may be different for different words but f; is xed. Let VVJ- be the number of words that appear exactly 3' times in novel A. Show that the expected number of distinct words appearing in novel B but not in novel A is IE[W1] E[W2]+E[W31E[W4]+m