Question

You’re a Seahawks fan, and the team is six weeks into its season. The number touchdowns scored in each game so far are given below: [1, 3, 3, 0, 1, 5]. Let’s call these scores x₁, . . . , x₆. Based on your data, you’d like to build a model to understand how many touchdowns the Seahaws are likely to score in their next game. You decide to model the number of touchdowns scored per game using a Poisson distribution. . So, for example, if λ = 1.5, then the probability that the Seahawks score 2 touchdowns in their next game is e −1.5 × 1.5 2 2! ≈ 0.25. To check your understanding of the Poisson, make sure you have a sense of whether raising λ will mean more touchdowns in general, or fewer.

1. Derive an expression for the maximum-likelihood estimate of the parameter λ governing the Poisson distribution, in terms of your touchdown counts x1, . . . , x6. (Hint: remember that the log of the likelihood has the same maximum as the likelihood function itself.)

2. Given the touchdown counts, what is your numerical estimate of λ?

Answer 1

The log-likelihood function is

b)

= mean =sum [1, 3, 3, 0, 1, 5]/6 = (1 + 3 + 3 + 1 + 5)/6 = 2.16666

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