Question

Use a conditioning with compound rv, if applicable. Please provide as much details as you can. Thanks in advance

You are invited to a party. Suppose the times at which invitees arrive are independent and distributed uniformly across the duration of the party. Suppose that, aside from yourself, the number of other people who are invited, N, is a Poisson random variable with mean λ . Let X represent the number of people who arrive before you.

(a) Find E[X].

(b) Find the probability that you are the last person to arrive

Answer 1

Date: --19/02/2019 thenumber of people arrive before t 0 otherwise Let X be the number of people that arrive to the party Let us consider the total number of people arrive to the party be V.including you, are invited to a party Given that I arrive at time t, the probability that any other person arrives the people arrive at the party are independent uniform (0,1 before me is precisely t Now, there are (N-1) people, each having a chance t of arriving before me, and doing this independently Thus, N, the number of people arriving before me is Binomial (N-1).) Thus, from standard formula B(x)-(N-1)

b) Given that, MT, and X-X, you are the nth person to arrive if exactly n-1 persons arive before you. Therefore, the probability that you are the nth person to arrive is P {Suppose you are the nth person person to arrive! X-X,M케흥 Pn Since X is aUniform (0,1) random variable, it follows that n-1) T(m+2 T(n)r(m-n+2) Since the integrand in the last term of the above equation is the p.d.f of a beta distributed random variable with parameters α-n and β-n-m +2 Hence the integral is equal to 1. Therefore we have to note that,「(t) = (t-1)!. If t is a positive integer. Since m and n are integers andm>n, we have m+1