Stochastic Process, Poisson Process

[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).

 

[mighty2000]

Hi there! I would be happy to help you with your homework. I specialize in analyzing and interpreting data, so I am familiar with the concepts of Poisson processes and binomial tests.

To answer your first question, we can use the formula E(Nt) = Var(Nt) = λt, where λ is the rate of arrivals and t is the time interval. In this case, we know that λ = 1/4 per minute, so E(NT) = Var(NT) = (1/4)(10) = 2.5. This means that on average, there will be 2.5 passengers waiting for the next bus.

For part a), we need to find E(NT | T), which is the expected number of passengers waiting given the arrival time of the next bus. Since Nt and T are independent, we can use the formula E(NT | T) = E(NT) = 2.5.

Moving on to the second question, we can use the formula E(T) = 1/λ and Var(T) = 1/λ^2 to find the expected and variance of T. So, E(T) = 1/λ = 4 minutes and Var(T) = 1/λ^2 = 16 minutes^2.

For part b), we need to find E(T | K), which is the expected time of failure given the number of shocks experienced. Since the probability of failure is p regardless of the number of shocks, we can use the formula E(T | K) = E(T) = 4 minutes.

I hope this helps you with your homework. Let me know if you have any further questions or need clarification on any of the concepts. Good luck!