Solve Both

Exercise 34 (#1.98). Let Py be a discrete distribution on {0,1,2. ... } and given Y = y. the conditional distribution of X be the binomial distri- bution with size y and probability p. Show that (i) if Y has the Poisson distribution with mean 0, then the marginal distri- bution of X is the Poisson distribution with mean po; (ii) if y + r has the negative binomial distribution with size r and proba- bility 7, then the marginal distribution of X +r is the negative binomial distribution with size r and probability 7/1 - (1 - p) (1 7). Exercise 35 (#1.85). Let X and Y be integrable random variables on the probability space (12. F. P) and A be a sub-o-field of F. Show that (i) if X