General prob-ability rules, conditional probability and Bayes theorem

Q1:5 Let Xi be Poisson distributed with parameter i for each i = 1; : : : ;m, and

suppose X1; : : : ;Xm are independent. Find the distribution of X1 + + Xm by the mgf.

(Hint: Compute the mgf of X1 + + Xm).

Q2：In a factory, there is a certain equipment which needs to be replaced immediately

after it goes bad. Let X1 and X2 denote the rst two times (in days) when the equipment

is replaced. The joint density of X1 and X2 is given by

fX1;X2(x1; x2) = e?x2 ; 0 x1 x2:

(a) Compute the pdf of the lifetime of the equipment (i.e., the marginal pdf of X1).

(b) Suppose that the second replacement happened after 15 days. Given that, what is the

conditional distribution of the time of the rst replacement? (Hint: Find X1jX2 = 15.)