The distribution of accidents for 84 randomly selected policies is as follows:

Number of Accidents	Number of Policies
0	32
1	26
2	12
3	7
4	4
5	2
6	1
Total	84

Determine which of the following models best represents these data.

• Negative binomial
• Discrete uniform
• Poisson
• Binomial
• Zero-modified Poisson

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

Key: A

[[math]]\begin{array}{cc}k & k n_{k} / n_{k-1} \\ 0 & \\ 1 & 0.81 \\ 2 & 0.92 \\ 3 & 1.75 \\ 4 & 2.29 \\ 5 & 2.50 \\ 6 & 3.00\end{array}[[/math]]

Positive slope implies that the negative binomial distribution is a good choice. Alternatively, the sample mean and variance are 1.2262 and 1.9131 respectively. With the variance substantially exceeding the mean, the negative binomial model is again supported.

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