To compute the probability of winning in a sudden death overtime, the important statistic is not goals per game, but time until the first goal. The assumption that goal-scoring is a Poisson process implies that the time between goals is exponentially distributed.
lam, we can compute the
time between goals like this:
lam = 3.4 time_dist = thinkbayes.MakeExponentialPmf(lam, high=2, n=101)
high is the upper bound of the
distribution. In this case I chose 2, because the probability of going
more than two games without scoring is small.
n is the number of values in the Pmf.
If we know
lam exactly, that’s
all there is to it. But we don’t; instead we have a posterior distribution
of possible values. So as we did with the distribution of goals, we make a
meta-Pmf and compute a mixture of Pmfs.
def MakeGoalTimePmf(suite): metapmf = thinkbayes.Pmf() for lam, prob in suite.Items(): pmf = thinkbayes.MakeExponentialPmf(lam, high=2, n=2001) metapmf.Set(pmf, prob) mix = thinkbayes.MakeMixture(metapmf) return mix
Figure 7-3 shows the resulting distributions. For time values less than one period (one third of a game), the Bruins are more likely to score. The time until the Canucks score is more likely to be longer.
I set the number of values,
n, fairly high in order to minimize the number of ties, since it is not possible for ...