An employer provides disability benefits to its employees for work-related and other injuries. The random variables [math]X[/math] and [math]Y[/math] denote the employer’s annual expenditures for work-related and other injuries, respectively. An actuarial study reveals the following information about [math]X[/math] and [math]Y[/math]:

1. The density of [math]X[/math] is [math]f(x)= \frac{1}{20 \sqrt{5}} e^{-x/(20 \sqrt{5})}, \, x \gt 0[/math]
2. [math]\operatorname{Var}(Y)=12500 [/math]
3. The correlation between [math]X[/math] and [math]Y[/math] is 0.20.

Calculate the variance of the employer’s total expenditures for work-related and other injuries.

• 12,500
• 13,500
• 15,500
• 16,500
• 18,972

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

Solution: D

X follows an exponential distribution with mean [math]20 \sqrt{5}[/math] and variance [math](20 \sqrt{5})^2 = 2000.[/math] Then,

[[math]] \operatorname{Cov} [ X , Y ] = \operatorname{Var} [ X ] = \operatorname{Var} [Y ] \operatorname{Corr} [ X , Y ] = \sqrt{2000} \sqrt{12500} ( 0.2 ) = 1000 . [[/math]]

It follows that

[[math]] \operatorname{Var} [ X + Y ]= \operatorname{Var} [ X ] + \operatorname{Var} [Y ] + 2\operatorname{Cov} [ X , Y ]= 2000 + 12,500 + 2 (1000 )= 16,500 . [[/math]]

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