Poisson Distribution Problem

Assume that when the German plays Hungary in soccer, each team scores independently as a homogeneous Poisson process with rates λ_{Germany }= 1 and λ_{Hungar}_{y }= 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. `

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)

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) = (λ_{1} / λ_{1} + λ_{2) }* 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?

Re: Poisson Distribution Problem

I think you have to redefine the rates on a per half rather than per game basis. I.e

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