For a discrete probability distribution, you are given the recursion relation

Calculate [math]p(4)[/math].

• 0.07
• 0.08
• 0.09
• 0.10
• 0.11

The distribution of accidents for 84 randomly selected policies is as follows:

Determine which of the following models best represents these data.

• Negative binomial
• Discrete uniform
• Poisson
• Binomial
• Zero-modified Poisson

Prescription drug losses, S, are modeled assuming the number of claims has a geometric distribution with mean 4, and the amount of each prescription is 40.

Calculate [math]\operatorname{E}[(S-100)_{+}][/math]

• 60
• 82
• 92
• 114
• 146

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

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

A risk has a loss amount that has a Poisson distribution with mean 3. An insurance policy covers the risk with an ordinary deductible of 2. An alternative insurance policy replaces the deductible with coinsurance [math]\alpha[/math], which is the proportion of the loss paid by the policy, so that the expected cost remains the same.

Calculate [math]\alpha [/math].

• 0.22
• 0.27
• 0.32
• 0.37
• 0.42

The distribution of the number of claims, N, is a member of the (a, b, 0) class. You are given:

1. [math]\operatorname{P}(N=k) = p_k[/math]
2. [math]\frac{p_6}{p_4} = 0.5 [/math] and [math]\frac{p_5}{p_4} = 0.8[/math]

A zero-modified distribution, [math]N^M[/math], associated with [math]N[/math] has [math]\operatorname{P}(N^M = 0) = 0.1 [/math].

Calculate [math]\operatorname{E}(N^M) [/math].

• 3.64
• 3.73
• 3.85
• 4.00
• 4.05

You are given the following properties of the distribution of the annual number of claims, N:

1. [math]\operatorname{P}(N=k) = p_k, \quad k = 0,1,2,\ldots[/math]
2. [math]p_0 = 0.45 [/math]
3. [math]\frac{p_n}{p_m} = \frac{m!}{n!} [/math] for [math]m \geq 1 [/math] and [math] n \geq 1 [/math]

Calculate the probability that at least two claims occur during a year.

• 0.16
• 0.18
• 0.21
• 0.23
• 0.26