# Stochastic Process, Poisson Process

1. Nov 27, 2011

### Synthemesc90

Hi, I need some help with this hw

1. Suppose that the passengers of a bus line arrive according to a Poisson process Nt with a rate of λ = 1 / 4 per minute. A bus left at time t = 0 while waiting passengers. Let T be the arrival time of the next bus. Then the number of passengers who are waiting for is NT. Suppose that T is random with uniform density in the range (9, 11), T ^ U (9,11), and also Nt and T are independent.
a) Find E (NT | T)
b) Find E (NT), Var (NT)

In PP E(Nt)=Var(Nt)=λt

2. A machine is subjected to shocks that occur according to Poisson process Nt with rate λ . The machine can suffer a failure due to one of these shocks, and the probability of a crash caused the fault is p, regardless of the number of previous shocks (shocks form a sequence of binomial test). Denote by K the total number of shocks experienced by the machine before going down, and let T be the time when the fault occurs.
a) Find E (T), Var (T).
b) Find E (T | K).