# Question

Suppose that W, the amount of moisture in the air on a given day, is a gamma random variable with parameters (t, Î²). That is, its density is f(w) = Î²eâˆ’Î²w(Î²w)tâˆ’1 / Î“(t), w > 0. Suppose also that given that W = w, the number of accidents during that dayâ€”call it Nâ€”has a Poisson distribution with mean w. Show that the conditional distribution of W given that N = n is the gamma distribution with parameters (t + n, Î² + 1).

## Answer to relevant Questions

Let W be a gamma random variable with parameters (t, Î²), and suppose that conditional on W = w, X1, X2, . . . ,Xn are independent exponential random variables with rate w. Show that the conditional distribution of W given ...Establish Equation (6.2) by differentiating Equation (6.4). If X and Y are jointly continuous with joint density function fX,Y(x, y), show that X + Y is continuous with density function Let A1, A2, . . . ,An be arbitrary events, and define Ck = {at least k of the Ai occur}. Show that Let X denote the number of the Ai that occur. Show that both sides of the preceding equation are equal to E[X]. A set of 1000 cards numbered 1 through 1000 is randomly distributed among 1000 people with each receiving one card. Compute the expected number of cards that are given to people whose age matches the number on the card.Post your question

0