What is the Probability of 2 Events Occurring in a Poisson Process?

[dargar]

Homework Statement

Events X, Y, Z are all Poisson processes. Event X has a rate of 1 per unit time , event Y has a rate of 2 per unit time and event Z has a rate of 3 per unit time.

Find the probability that 2 events (of any type) occur during the interval (0, 3).

Homework Equations

Maybe this is relevant
P(N=k) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}

The Attempt at a Solution

So \lambda_X = 1, \lambda_Y = 2 and \lambda_Z = 4. Also k = 2 and t =3.

Is it correct to think of it as say A = X \cup Y \cup Z. Then the answer is:

P(N=2) = \frac{(7(3))^2 e^{-7(3)}}{2!} where \lambda_A = 1 + 2 + 4 = 7.

 

[korican04]

Homework Statement

Events X, Y, Z are all Poisson processes. Event X has a rate of 1 per unit time , event Y has a rate of 2 per unit time and event Z has a rate of 3 per unit time.

Find the probability that 2 events (of any type) occur during the interval (0, 3).

Homework Equations

Maybe this is relevant
P(N=k) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}

The Attempt at a Solution

So \lambda_X = 1, \lambda_Y = 2 and \lambda_Z = 4. Also k = 2 and t =3.

Is it correct to think of it as say A = X \cup Y \cup Z. Then the answer is:

P(N=2) = \frac{(7(3))^2 e^{-7(3)}}{2!} where \lambda_A = 1 + 2 + 4 = 7.

I believe that this is correct. If X Y and Z are independent then a random variable say A=X+Y+Z would have a poisson distribution with rate of \lambda_X +\lambda_Y+\lambda_Z
Although you initially wrote \lambda_Z =3 ,but put down 4.