Poisson processes

Question 1

Define the mean value function of a nonhomogeneous Poisson process with intensity function λ (t)

Answer 1

m(t) = integral_0^t λ(s)ds

Question 2

State the conditions for a counting process to be a nonhomogeneous Poisson process with intensity function λ (t)

Answer 2

N(0) = 0
{N(t), t>=0} has independent increments
P(N(t+h) - N(t) = 1) = λ(t)h + o(h)
P(N(t+h) - N(t) >= 2) = o(h)

Question 3

What is a counting process?

Answer 3

counts event up to time t {N_t}_(t>=0) 
N_0 = 0
N_t is in the naturals_0
s N_s <= N_t (the # of events in (s,t] )
we have piecewise jumps of size 1

Question 4

State the distribution of the time to an nth event of a Poisson process. Why is this?

Answer 4

Gamma(n,λ)

The inter arrival times are exponentially distributed with parameter λ, and the sum of these is Gamma.

Question 5

Suppose {N_t^i } ∼ Poi(λ_i) and set N_t to be the sum of all N_t^i. What is the distribution of N_t?

Answer 5

N_t is a Poisson process with rate λ = Sum_{i=1}^n λi

pg 83

Question 6

Define a compound Poisson process

State its expectation and variance

Answer 6

N_t Poisson with rate λ and Y_i i.i.d. and independent of N_t. Then S_t = sum_{i=1}^{N_t} Y_i is a compound Poisson process.

E(St) = λtE(Y_1)
Var(St) = λtE(Y_1 ** 2).

pg 88

Question 7

What is a PMF? State the PMF of a Poisson(λ) distribution.

Answer 7

P(N_t = k) = [ (λt)^k / k! ] exp(-λt)

Question 8

What is the distribution of the inter arrival times of a Poisson(λ) process?

Answer 8

Exp(λ)