1. ## Likelihood Question

Hi everyone,
I'm working through some max likelihood questions and am badly stuck on this one. Please could you take a look at what I'm doing and tell me if I'm going in the right direction?

Q. Team A and B play two games of soccer, each game having two halves of equal length. Assume the number of goals are independent of the other team's score and of their own score in the other half. The scores are as follows:

A B
4-1, with 2-0 at half time
3-1, with 1-1 at half time

Suppose A score at a rate 2α in the first half and at rate α in the second half and B score at a rate β in the first half and at rate 2β in the second half.

(i) Obtain the liklihood of the match results given about under this model.
(ii) Show that the estimates of α and β depend only on the total number of goals scored by a team and thus that the halftime scores are irrelevant.
(iii) Obtain values for α and β.

--
Solution so far:

First consider team A.
For the first half of A's games, λ = 2α
For the second half of A's games, λ = α

The two halves are independent, so we can combine the Poissons to get: (e^-3α)((3α)^x)/x!

but if we take the pdfs of the two halves and combine by multiplying, we get:

(e^-2α)((2α)^x)/x!*(e^-α)((α)^x)/x!

which is not the same...

I'm fairly certain what I'm doing so far is incorrect. Is this even the right model? Am I correct in substituting the α and 2α for the λ?
I would really appreciate any pointers or tips in the right direction.

Thanks!

2. Originally Posted by languagelearner23
Hi everyone,
I'm working through some max likelihood questions and am badly stuck on this one. Please could you take a look at what I'm doing and tell me if I'm going in the right direction?

Q. Team A and B play two games of soccer, each game having two halves of equal length. Assume the number of goals are independent of the other team's score and of their own score in the other half. The scores are as follows:

A B
4-1, with 2-0 at half time
3-1, with 1-1 at half time

Suppose A score at a rate 2α in the first half and at rate α in the second half and B score at a rate β in the first half and at rate 2β in the second half.

(i) Obtain the liklihood of the match results given about under this model.
(ii) Show that the estimates of α and β depend only on the total number of goals scored by a team and thus that the halftime scores are irrelevant.
(iii) Obtain values for α and β.

--
Solution so far:

First consider team A.
For the first half of A's games, λ = 2α
For the second half of A's games, λ = α

The two halves are independent, so we can combine the Poissons to get: (e^-3α)((3α)^x)/x!

but if we take the pdfs of the two halves and combine by multiplying, we get:

(e^-2α)((2α)^x)/x!*(e^-α)((α)^x)/x!

which is not the same...

I'm fairly certain what I'm doing so far is incorrect. Is this even the right model? Am I correct in substituting the α and 2α for the λ?
I would really appreciate any pointers or tips in the right direction.

Thanks!
You have four data points the scores in each half of each or the games, these are (2,0), (2,1) from game 1, and (1,1), (2,0) from the second game.

The likelihood is the product of the likelihoods for each of the four halves (where (2,0) and (1,1) are first halves and (2,1) and (2,0) are second halves).

CB