For Problems 1-3, take a Poisson process N = (N(t), t greaterthanorequalto 0) wi

Question

For Problems 1-3, take a Poisson process N = (N(t), t greaterthanorequalto 0) with intensity lambda = 4. Problem 1. Find the distribution of N(5) - N(2), its expectation and variance. Problem 2. Find the distribution of T_5 - T_2, its expectation and variance. Problem 3. Find P(N(5) = 6 | N(3) = 1). Problem 4. Show that (N(3t), t greaterthanorequalto 0) is also a Poisson process, and find its intensity. Problem 5. Take two independent Poisson processes N_1 = (N_1 (t), t greaterthanorequalto 0) and N_2 (N_2 (t), t greaterthanorequalto 0), with intensities lambda_1 = 3 and lambda_2 = 4. Show that N(t) = N_1(2t) + N_2(3 t) is a Poisson process, and find its intensity.

Explanation / Answer

Problem 1:

As the mean and variance of a poisson distribution are equal we get:

E(N(t)) = Var(N(t) ) = 4t

Therefore now let K = N(5) - N(2)

Then we get:

E(K) = E( N(5) - N(2) ) = 5*4 - 2*4 = 12

Similarly variance is computed as:

Var(K) = Var( N(5) - N(2) ) = Var( N(5) ) + Var( N(2) ) = 5*4 + 2*4 = 28

Therefore the mean for N(5) - N(2) is 12 and the variance is 28

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.