Math 419
Common Discrete Random Variables
Problem Set 4
1. A probability distribution of claim sizes for a home insurance policy is given in the table below :
Claim Size
5
10
15
20
25
30
35
40
Probability
0.01
0.1
0.09
0.21
0.17
0.25
0.1
0.07
What percentage of claims are within 1.5 standard deviations of the mean claim size?
2. Bob buys 14 securities of Company Y. Securities of Company X are the same price, but the variance
of the price of X is 30 and the variance of the price of Y is 35. According to market research, the
price of X goes up when the price of Y goes down, with a covariance of –15. How many securities
of X should be purchased to minimize the variance of Bob's portfolio?
3. The Acme company has a death benefit f 100 for every employee. Acme has insurance that pays the
total death benefit if it is over 400 in a year. The number of employees that die during the year is
Poisson with mean 2. Calculate the expected annual cost to the company of providing the death
benefit, disregarding the cost of the insurance.
4. The total number of accidents in the months of May, June, and July is modeled with a Poisson
random variable with mean 1.5. The rate of accidents is assumed to not vary during the three month
period.
The total number of accidents in the remainder of the year is also modeled by a Poisson random
variable, this time with mean 0.5. Again, the rate of accidents is assumed to not vary during the nine
month period.
The number of accidents in any month is independent from the number of accidents in any other
month. What is the probability that in the months of July and August, there is a total of 2 accidents?
5. A manufacturer sells boxes of 50 industrial fuses at a price of 300 per box. Each fuse has a 1%
chance of being faulty, independent of all other fuses. The manufacturer promises to refund 100 to
customers for each faulty fuse in a box, up to a maximum of 300 per box.
The manufacturer sells 1,000 boxes of fuses. Calculate the expected total refund.
6. The probability that a person carries a certain gene is 0.2. In a trial, a doctor tests people for gene and
counts how many she tests before she finds one with the gene. These trials are conducted at ten
major city hospitals.
Funding for research into the gene will be made available if in at least 9 of the trials it takes fewer
than four people without the gene before the first with the gene.
Determine the probability that funding is not made available.
Math 419
Common Discrete Random Variables
Problem Set 4
7. A printing firm operates vans to deliver its products. The probability that a van needs repair in a
given month is one-third. The probability that a van needs repair is independent from one month to
the next and from one van to another. Calculate the probability that at least three months pass without
the need for repairs before the third month in which repairs are required.
8. The number of claims received per week at an insurance company has a Poisson distribution. The
probability that four claims are received in a week is half the probability that three claims are
received in a week. Calculate the probability that the number of claims received in a week exceed
two.
9. The number of accidents at an intersection per year is described by a Poisson random variable with
mean 8. Residents have campaigned for a safety device to be installed. The intersection is monitored
for the next nine months, and it is agreed that if there is at least one month with two or more
accidents, then the device will be installed. What is the probability that the safety device will be
installed?
10. The number of months a system will function is described by N, which has probability mass
function
P(N = n) = (150n) / (151n+1) , n = 0, 1, 2,....
A system has been functioning for 11 months. What is the expected number of months remaining
before the system fails?