Independent Poisson Processes Word Problem

[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.

Homework Statement

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.

Homework Equations

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?

 

[Ray Vickson]

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.

Homework Statement

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.

Homework Equations

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?

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

RGV