You are given:

1. The number of claims has probability function:

[[math]] p(x) = \binom{m}{x}q^x(1-q)^{m-x}, \, x = 0,1,\ldots,m [[/math]]

1. The actual number of claims must be within 1% of the expected number of claims with probability 0.95.
2. The expected number of claims for full credibility is 34,574.

Calculate q.

• 0.05
• 0.10
• 0.20
• 0.40
• 0.80

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

Key: B

Let n be the number of observations. For full credibility,

[[math]] \begin{aligned} n = \left(\frac{1.96}{0.01} \right)^2 \frac{mq(1-q)}{(mq)^2} = 38,416 \frac{1-q}{mq}. \end{aligned} [[/math]]

The required expected number of claims is

[[math]] nmq = 34,574 = 38, 416 \frac{1-q}{mq} mq = 38,416(1-q). [[/math]]

Then q = 1 – 34,574/38,416 = 0.1.

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