A discrete probability distribution has the following properties:

1. [math]p_k = c(1 + \frac{1}{k})p_{k+1} [/math] for [math]k = 1, 2, \ldots [/math]
2. [math]p_0 = 0.5 [/math]

Calculate c.

• 0.06
• 0.13
• 0.29
• 0.35
• 0.40

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

Key: C

Write (i) as [math]\frac{p_k}{p_{k-1}} = c + \frac{c}{k}[/math]

This is an (a, b, 0) distribution with a = b = c. Which?

1. If Poisson, a = 0, so b = 0 and thus [math]p_0 = 0.5[/math] and [math]p1 = p2 = \cdots = 0[/math] . The probabilities do not sum to 1 and so not Poisson.
2. If Geometric, b = 0, so a = 0. By same reasoning as #1, not Geometric.
3. If binomial, a and b have opposite signs. But here a = b, so not binomial.
4. Thus negative binomial.

[[math]] \begin{aligned} &1 = \frac{a}{b} = \frac{\beta/(1+\beta)}{(r-1)\beta/(1-\beta)} = \frac{1}{r-1} \, \textrm{so } r = 2 \\ &p_0 = 0.5 = (1 + \beta)^{-r} = (1 + \beta )^{-2} \Rightarrow \beta = \sqrt{2} − 1 = 0.414 \\ &c = a = \beta / (1 + \beta ) = 0.29 \end{aligned} [[/math]]

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