New dental and medical plan options will be offered to state employees next year. An actuary uses the following density function to model the joint distribution of the proportion [math]X[/math] of state employees who will choose Dental Option 1 and the proportion [math]Y[/math] who will choose Medical Option 1 under the new plan options:

[[math]] f(x,y) = \begin{cases} 0.50, \,\, 0 \lt x \lt 0.5, 0 \lt y \lt 0.5 \\ 1.25, \,\, 0 \lt x \lt 0.5, 0.5 \lt y \lt 1 \\ 1.50, \,\, 0.5 \lt x \lt 1, 0 \lt y \lt 0.5 \\ 0.75, \,\, 0.5 \lt x \lt 1, 0.5 \lt y \lt 1 \\ \end{cases} [[/math]]

Calculate [math]\operatorname{Var} (Y | X = 0.75)[/math].

• 0.000
• 0.061
• 0.076
• 0.083
• 0.141

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

Solution: C

[[math]] f(y | x = 0.75) = \frac{f(0.75,y)}{\int_0^1 f(0.75,y) dy} = \frac{f(0.75,y)}{1.125}. [[/math]]

Thus,

[[math]] f(y | x= 0.75) = \begin{cases} 4/3, \, 0 \lt y \lt 0.5 \\ 2/3, \, 0.5 \lt y \lt 1 \end{cases} [[/math]]

which leads to [math]\operatorname{Var}(Y | X = 0.75 ) = 11/144 = 0.076 [/math].

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