please answer the folloowing

8. Given 1/, A > 0 and p E (0,1), suppose - X0 ~ Poise), 0 Z N P0is()\\), o and for each is 2 0, let Yk ~ binomUc, p) With the convention Y0 = 0 (if there is no trial, then the number of successful trials must be 0). Assume further that X0, Yk, Z, k = 0, 1, . .. are all independent. (a) Let X1 be a discrete RV such that for all 3, k 2 0, Find the distribution of X1. :1: HINT: Suppose the sample probability space is a disjoint union of events {Akgl and Y is a non negative RV. Then Em = ZElYIAQMAk). 121 (b) Generally, let X\