X is a discrete random variable with a probability function that is a member of the (a,b,0) class of distributions. You are given:

1. [math]\operatorname{P}(X = 0) = P(X = 1) = 0.25[/math]
2. [math]\operatorname{P}(X=2) = 0.1875[/math]

Calculate [math]\operatorname{P}(X = 3) [/math]

• 0.120
• 0.125
• 0.130
• 0.135
• 0.140

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

Key: B

[[math]] \begin{aligned} p_k &= (a + \frac{b}{k})p_{k-1} \\ 0.25 &= (a+b)0.25 \Rightarrow a = 1-b \\ 0.1875 &= (a + \frac{b}{2})(0.25) \Rightarrow 0.1875 = (1 − 0.5b)(0.25) \Rightarrow b = 0.5, a = 0.5 \\ p_3 &= (0.5 + \frac{0.5}{3})(0.1875) = 0.125 \end{aligned} [[/math]]

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