You are given:

1. The distribution of the number of claims per policy during a one-year period for 10,000 insurance policies is:

   Number of Claims per Policy	Number of Policies
   0	5000
   1	5000
   2 or more	0

2. You fit a binomial model with parameters m and q using the method of maximum likelihood.

Calculate the maximum value of the loglikelihood function when [math]m = 2[/math].

• −10,397
• −7,781
• −7,750
• −6,931
• −6,730

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key: B

[[math]] \begin{aligned} L(q) &= \left [ \binom{2}{0}(1-q)^2\right]^{5000} \left[ \binom{2}{1}q(1-q)\right]^{5000} = 2^{5000}q^{5000}(1-q)^{15000} \\ l(q) &= 5000 \ln(2) + 5000 \ln(q) + 15000 \ln(1 − q) \\ l^{'}(q) &= 5000q^{−1} − 15000(1 − q)^{−1} = 0 \\ \hat{q} &= 0.25 \\ l(0.25) &= 5000 \ln(2) + 5000 \ln(0.25) + 15000 \ln(0.75) \\ &= −7780.97. \end{aligned} [[/math]]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.