The time, [math]T[/math], that a manufacturing system is out of operation has cumulative distribution function

[[math]] F(t) = \begin{cases} 1-(2/t)^2, \, t \gt 2 \\ 0, \, \textrm{Otherwise.} \end{cases} [[/math]]

The resulting cost to the company is [math]Y = T^2[/math]. Let [math]g[/math] be the density function for [math]Y[/math]. Determine [math]g(y)[/math], for [math]y \gt 4 [/math].

• [math]\frac{4}{y^2}[/math]
• [math]\frac{8}{y^{3/2}}[/math]
• [math]\frac{8}{y^3}[/math]
• [math]\frac{16}{y}[/math]
• [math]\frac{1024}{y^5}[/math]

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

Solution: A

The distribution function of Y is given by

[[math]] G(y) = Pr( T^2 \leq y ) = Pr( T \leq \sqrt{y} ) = F( \sqrt{y} ) = 1-4/y [[/math]]

for [math] y \gt 4 [/math]. Differentiate to obtain the density function [math]g(y) = 4y^{-2}[/math].

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