# Poisson Distribution Problem

• Feb 15th 2014, 04:45 PM
Rorschach27
Poisson Distribution Problem
Assume that when the German plays Hungary in soccer, each team scores independently as a homogeneous Poisson process with rates λGermany = 1 and λHungary = 3 goals per game.

a) Expected number of total goals in a single game.

b) Expected number of total goals scored in the first half, given that 2 total goals were scored in the entire game.

c) Expected number of goals Germany scored in the entire game, given that 2 total goals were scored in the first half.

What I really need help with is getting the formulas for parts b & c. I'm having difficulty forming what the Expectation formula should be. `
• Feb 15th 2014, 09:52 PM
chiro
Re: Poisson Distribution Problem
Hey Rorshach27.

Hint: If two process are independent then P(A = a, B = b) = P(A = a) * P(B = b) and P(A|B) = P(A = a,B = b)/P(B=b)
• Feb 16th 2014, 11:32 AM
Rorschach27
Re: Poisson Distribution Problem
Thanks for the help,

at this point I'm still having trouble with part c of this problem.

I've come to the conclusion that if I use the conditional expectation of

E(X|X + Y = n) = (λ1 / λ1 + λ2) * n
I might get to the answer, however I'm not sure how to account for the fact that our given statement states two goals were scored in the first half

Any help?
• Feb 16th 2014, 12:11 PM
romsek
Re: Poisson Distribution Problem
I think you have to redefine the rates on a per half rather than per game basis. I.e

$$\lambda_{Germany}=0.5 \mbox{ and }\lambda_{Hungary}=1.5$$