You are given:

1. Losses follow an exponential distribution with mean [math]\theta[/math] .
2. A random sample of 20 losses is distributed as follows:

Loss Range	Frequency
[0, 1000]	7
(1000, 2000]	6
(2000, [math]\infty[/math])	7

Calculate the maximum likelihood estimate of [math]\theta[/math].

• Less than 1950
• At least 1950, but less than 2100
• At least 2100, but less than 2250
• At least 2250, but less than 2400
• At least 2400

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

Key: B

[[math]]\begin{aligned} & L=F(1000)^{7}[F(2000)-F(1000)]^{6}[1-F(2000)]^{7} \\ & =\left(1-e^{-1000 / \theta}\right)^{7}\left(e^{-1000 / \theta}-e^{-2000 / \theta}\right)^{6}\left(e^{-2000 / \theta}\right)^{7} \\ & =(1-p)^{7}\left(p-p^{2}\right)^{6}\left(p^{2}\right)^{7}=p^{20}(1-p)^{13} \end{aligned}[[/math]]

where [math]p=e^{-1000 / \theta}[/math]. The maximum occurs at [math]p=20 / 33[/math] and so [math]\hat{\theta}=-1000 / \ln (20 / 33)=1996.90[/math].

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