The monthly profit of Company I can be modeled by a continuous random variable with density function [math]f[/math]. Company II has a monthly profit that is twice that of Company I. Let [math]g[/math] be the density function for the distribution of the monthly profit of Company II.

Determine [math]g(y)[/math] where it is not zero.

• [math]\frac{1}{2}f(\frac{y}{2})[/math]
• [math]f(\frac{y}{2})[/math]
• [math]2f(\frac{y}{2})[/math]
• [math]2f(y)[/math]
• [math]2f(2y)[/math]

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

Solution: A

Let X and Y be the monthly profits of Company I and Company II, respectively. We are given that the pdf of X is f . Let us also take g to be the pdf of Y and take F and G to be the distribution functions corresponding to f and g . Then

[[math]]G(y) = \operatorname{P}[Y ≤ y] = \operatorname{P}[2X ≤ y] = \operatorname{P}[X ≤ y/2] = F(y/2)[[/math]]

and

[[math]]g(y) = G^{'}(y) = d/dy F(y/2) =1/2 F^{′}(y/2) =1/2 f(y/2) .[[/math]]

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