Losses due to burglary are exponentially distributed with mean 100. The probability that a loss is between 40 and 50 equals the probability that a loss is between 60 and [math]r[/math], with [math]r \gt 60[/math]. Calculate [math]r[/math].

• 68.26
• 70.00
• 70.51
• 72.36
• 75.00

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

Solution: B

Let [math]X[/math] be the number of burglaries. Then,

[[math]] \begin{align*} \operatorname{E}(X | X \geq 2) = \frac{\sum_{x=2}^{\infty}xp(x)}{1-p(0)-p(1)} &= \frac{\sum_{x=0}^{\infty}xp(x)-(0)p(0)-(1)p(1)}{1-p(0)-p(1)} \\ &= \frac{1-p(1)}{1-p(0)-p(1)} = \frac{1-e^{-1}}{1-e^{-1}-e^{-1}} = 2.39. \end{align*} [[/math]]

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