# Think Bayes Chapter 7: Prediction

Page 72, Think Bayes by Allen B. Downey.

Author introduces the Boston Bruins problem where we are supposed to determine the posterior distribution of λ, where λ is the long-term average goals per game.

In section 7.2 he proceeds to compute the distribution of goals for each team.

I have trouble understanding why this is done. Are these two not the same?

The first plot seems to be confusingly labeled. It is the posterior distribution of the parameter of interest $$\lambda$$. So if the model is that the number of goals is Poisson distributed, the first plot shows the computed distribution of $$\lambda$$ after looking at the data. That is: given the data, what are likely values of $$\lambda$$ for future matches? It seems that bruins will, on average, score higher than the canucks.
The second plot is for some fixed lambda. For instance, after sampling once from the distribution of $$\lambda$$ above, a number 2.7 is drawn for the "bruins" and 2.9 for the canucks. This gives a specific shape of the density of a Poisson distribution with that parameter. That is a specific probability of goals in a single match with that parameter. Note how the slightly higher number for the canucks shifts the pmf to the right. If you look at the area under the curves you see that it is less likely that they score less than three goals, and more that they score 3 or more goals than the bruins.
The confusion arises because the expected value (mean) of a random variable distributed as Poisson, is equal to the parameter $$\lambda$$.