# Independent Poisson Processes Word Problem

1. Nov 14, 2012

### Ocifer

Hello, hopefully this is the right place. This is a homework question, so it should definitely be in this forum, but I wasn't sure which sub-forum to put this rather elementary stats question.

1. The problem statement, all variables and given/known data
In my introductory mathematical statistics class, we've been given the following word problem having to do with Poisson processes.

We are to consider a football game, where each team's score follows its own Poisson process. So there are two teams, team A and team B and, each team's final score has its own lambda parameter:

Team A scores $\lambda_A$ goals per game.
Team B scores $\lambda_B$ goals per game.

What is the probability of team B winning over team A, with a final score of 3-2.

2. Relevant equations
3. The attempt at a solution
My intuition (which I would like to confirm/have critiqued) is that since the processes are independent of each other I should find the following.

Let $X_A$ be the final score of team A
Let $X_B$ be the final score of team B

$Pr(X_A = 2) = \lambda_A ^ 2 e^{- \lambda_A } / 2!$
$Pr(X_B = 2) = \lambda_B ^ 3 e^{- \lambda_B } / 3!$

And then the probability that team B wins with a score of 3 to 2 is simply the product of the two probabilities above, namely:

$Pr(X_A = 2 \cap X_B = 3) = Pr(X_A = 2) \cdot Pr(X_B = 3)$

Is this the correct approach?

2. Nov 14, 2012

### Ray Vickson

Yes, it is correct (given the rather unlikely scenario that the two scores are independent).

RGV