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: D

The cumulative distribution function for the exponential distribution is

[[math]] F(x) = 1-e^{-\lambda x} = 1-e^{-x/\mu} = 1-e^{-x/100}, x \gt 0. [[/math]]

From the given probability data,

[[math]] \begin{align*} F(50) - F(40) &= F(r) - F(60) \\ 1-e^{-50/100} - (1-e^{-40/100}) &= 1-e^{-r/100} - (1-e^{-60/100}) \\ e^{-40/100} - e^{-50/100} &= e^{-60/100} - e^{-r/100} \\ e^{-r/100} &= e^{-60/100} - e^{-40/100} + e^{-50/100} = =0.4850 \\ -r/100 &= \ln(0.4850) = -0.7236 \\ r &= 72.36. \end{align*} [[/math]]

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