For an aggregate loss distribution S:

1. The number of claims has a negative binomial distribution with [math]r = 16[/math] and [math]\beta = 6[/math] .
2. The claim amounts are uniformly distributed on the interval (0, 8).
3. The number of claims and claim amounts are mutually independent.

Calculate the premium such that the probability that aggregate losses will exceed the premium is 5% using the normal approximation for aggregate losses.

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

Key: D

We have the following table:

Item	Dist	E()	Var()
Number claims	NB(16,6)	16(6) = 96	16(6)(7) = 672
Claim amounts	U(0,8)	(8 − 0)2 / 12 = 5.33
Aggregate		4(96) = 384	96(5.33) + 672(42 ) = 11, 264

Premium = [math]\operatorname{E}(S ) + 1.645 \sqrt{\operatorname{E}[S )} = 384 + 1.645 \sqrt{11,264} = 559 [/math]

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