Math 419
Continuous Random Variables
Problem Set 5
1. An insurance policy pays for a random loss X subject to a deductible of 0.3. The loss amount is
modeled as a continuous random variable with density function f(x) = 2x , 0 < x <1. Given a random
loss X, find the probability that the insurance payment is less than 0.5.
2. A person is 20 years old now. His lifetime from now is modeled via a probability density function
f(x) = ke – 0.01x
for x > 0. What is the probability that he lives to 40 years of age?
3. Suppose X is a continuous random variable with probability density function f(x) = 0.5e – x / 2. Find its
expected value.
4. Suppose a random variable X has a cumulative distribution function F(x) = 1 – (125/(x + 5)3 ), for
x > 0. Find its expected value.
5. Claim amounts are modeled by a random variable with cumulative distribution function
F(x) = 1 – e – x / 2000 , x > 0. If the policy has a benefit maximum of 3,000, what is the probability of
the insurance company paying exactly 3,000 on a given claim?
6. The remaining lifetime of a 70-year-old man is modeled by the random variable X with probability
density function
f(x) = k / (x + 5)2 , for 0 < x < 30
and zero otherwise (k is a positive constant). Calculate the probability that the man will live 5 years
and then die during the following 5 years.
7. An insurer's annual weather-related loss X is a random variable with density function
f(x) = (2.5 (200)2.5) / x 3.5 , for x > 200
and zero otherwise. Calculate the difference between the 30th and 70th percentiles.
8. An actuary models claim amounts using the probability density function
f(x) = (10 – x) / 50 , for 0 < x < 10,
where x is in thousands. For Policy 1, the payment for a claim size of x is x for 0 < x < 4, and 4 for
4 < x < 10. For Policy 2, the payment for a claim size of x is 0 for 0 < x < 4, and x – 4 for
4 < x < 10. Calculate the difference in expected payments for the two policies.
9.An insurance policy reimburses a loss up to a benefit limit of 10. The policy holder's loss X follows a
distribution with probability density function f(x) = 2 / x3 , for x > 1, and zero otherwise. What is the
median benefit paid under this policy?