Let [math]X[/math] be the percentage score on a college-entrance exam for students who did not participate in an exam-preparation seminar. [math]X[/math] is modeled by a uniform distribution on [a, 100].

Let [math]Y[/math] be the percentage score on a college-entrance exam for students who did participate in an exam-preparation seminar. [math]Y[/math] is modeled by a uniform distribution on [1.25a, 100].

It is given that [math]\operatorname{E}( X^2 ) = 19, 600. [/math]

Calculate the 80th percentile of [math]Y[/math].

• 0.64
• 0.74
• 0.85
• 0.87
• 0.94

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

Solution: E

Using the formulas for the variance and mean of the uniform distribution:

[[math]] \begin{align*} \operatorname{E}(X^2) = \operatorname{Var}((X) + \operatorname{E}(X)^2 &= \frac{(100-a)^2}{12} + \left(\frac{100+a}{2}\right)^2 \\ &= \frac{100^2 -200a + a^2 +3(100)^2 + 600a + 3a^2}{12} \\ &= \frac{40000 + 400a + 4a^2}{12} \\ &= \frac{19600}{3} \\ 0 &= 40000 -78400 + 400a + 4a^2 \\ 0 &= a^2 + 100a - 9600 \\ 0 &= (a-60)(a+160) \\ a &= 60. \end{align*} [[/math]]

Then, [math]Y[/math] is uniform on the interval 1.25(60) = 75 to 100. The 80^th percentile is 75 + 0.8(25) = 95.

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