The probability of [math]x[/math] losses occurring in year 1 is [math]0.5x[/math] for [math]x=0,1, 2,\ldots[/math]. The probability of [math]y[/math] losses in year 2 given [math]x[/math] losses in year 1 is given by the table:

Calculate the probability of exactly 2 losses in 2 years.

• 0.025
• 0.031
• 0.075
• 0.100
• 0.131

Individuals purchase both collision and liability insurance on their automobiles. The value of the insured’s automobile is V. Assume the loss L on an automobile claim is a random variable with cumulative distribution function

Calculate the probability that the loss on a randomly selected claim is greater than the value of the automobile.

• 0.00
• 0.10
• 0.25
• 0.75
• 0.90

For a certain insurance company, 10% of its policies are Type A, 50% are Type B, and 40% are Type C. The annual number of claims for an individual Type A, Type B, and Type C policy follow Poisson distributions with respective means 1, 2, and 10.

Let [math]X[/math] represent the annual number of claims of a randomly selected policy. Calculate the variance of [math]X[/math].

• 5.10
• 16.09
• 21.19
• 42.10
• 47.20

Find [math]E(X^Y)[/math], where [math]X[/math] and [math]Y[/math] are independent random variables which are uniform on [math][0, 1][/math].

• 0.6931
• 0.7131
• 0.7344
• 0.7544
• 0.775

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

The joint probability density for [math]X[/math] and [math]Y[/math] is

Calculate the variance of [math]Y[/math] given that [math]X \gt 3 [/math] and [math]Y \gt 3 [/math].

• 0.25
• 0.50
• 1.00
• 3.25
• 3.50

Two life insurance policies, each with a death benefit of 10,000 and a one-time premium of 500, are sold to a married couple, one for each person. The policies will expire at the end of the tenth year. The probability that only the wife will survive at least ten years is 0.025, the probability that only the husband will survive at least ten years is 0.01, and the probability that both of them will survive at least ten years is 0.96.

Calculate the expected excess of premiums over claims, given that the husband survives at least ten years.

• 350
• 385
• 397
• 870
• 897

The distribution of [math]Y[/math], given [math]X[/math] , is uniform on the interval [0, [math]X[/math]]. The marginal density of [math]X[/math] is

Determine the conditional density of [math]X[/math] , given [math]Y = y[/math] where positive.

• 1
• 2
• 2x
• 1/y
• 1/(1-y)

A policyholder has probability 0.7 of having no claims, 0.2 of having exactly one claim, and 0.1 of having exactly two claims. Claim amounts are uniformly distributed on the interval [0, 60] and are independent. The insurer covers 100% of each claim.

Calculate the probability that the total benefit paid to the policyholder is 48 or less.

• 0.320
• 0.400
• 0.800
• 0.892
• 0.924

A theme park conducts a study of families that visit the park during a year. The fraction of such families of size [math]m[/math] is [math]\frac{8-m}{28}, \, m = 1,\ldots,7[/math]

For a family of size [math]m[/math] that visits the park, the number of members of the family that ride the roller coaster follows a discrete uniform distribution on the set [math]\{1,\ldots, m\}[/math].

Calculate the probability that a family visiting the park has exactly six members, given that exactly five members of the family ride the roller coaster.

• 0.17
• 0.21
• 0.24
• 0.28
• 0.31

In a large population of patients, 20% have early stage cancer, 10% have advanced stage cancer, and the other 70% do not have cancer. Six patients from this population are randomly selected.

Calculate the expected number of selected patients with advanced stage cancer, given that at least one of the selected patients has early stage cancer.

• 0.403
• 0.500
• 0.547
• 0.600
• 0.625