A large university will begin a 13-day period during which students may register for that semester’s courses. Of those 13 days, the number of elapsed days before a randomly selected student registers has a continuous distribution with density function [math]f(t)[/math] that is symmetric about [math]t = 6.5 [/math] and proportional to [math]1/(t + 1)[/math] between days 0 and 6.5.

A student registers at the 60th percentile of this distribution.

Calculate the number of elapsed days in the registration period for this student.

• 4.01
• 7.80
• 8.99
• 10.22
• 10.51

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

Solution: C

Let [math]T[/math] be the time of registration. Due to symmetry of the density function about 6.5. The constant of proportionality, c, can be solved from

[[math]] 0.5 = \int_0^{0.5}c \frac{1}{t+1} dt = c \ln(t+1) \Big |_0^{6.5} = c\ln(7.5), [[/math]]

which gives [math]c = 0.5/\ln(7.5) [/math]. Again using the symmetry, if 60^th percentile of [math]T[/math] is at [math]k[/math], then [math]\operatorname{P}[T ≤ 13 − k ] = 0.4.[/math] Thus,

[[math]] \begin{align*} 0.4 &= \operatorname{P}[T \leq 13-k] = \int_0^{13-k} \frac{0.5}{\ln(7.5)} \frac{1}{t+1} dt = \frac{0.5}{\ln(7.5)} \ln(14-k) \\ \ln(14-k) &= 0.8\ln(7.5) = 1.6119 \\ 14-k &= e^{1.6119} = 5.0124 \\ k &= 8.99 \end{align*} [[/math]]

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