Download Exam 2 1. A family has two cars. The number of accidents in a year

Exam 2
1. A family has two cars. The number of accidents in a year for the first
is a binomial random variable with n = 3 and p = 0.1. The number
of accidents in a year for the second is a binomial random variable
with n = 2 and p = 0.05. These random variables are independent.
Find the probability that the family’s cars are involved in at least 2
accidents in a year.
A. 0.0535
B. 0.0879
C. 0.1649
D. 0.2164
E. 0.2343
2. The joint density for losses X and Y is 6/(x + y + 1)4 on 0 ≤ x, y.
Find the probability that X ≥ 3 and Y ≥ 2.
A. 1/120
B. 1/36
C. 7/48
D. 1/6
E. 7/8
3. The remaining lifetimes of a wife and husband have independent density functions 2/(x + 1)3, x > 0 and 3/(y + 1)4, y > 0, respectively.
What is the probability the wife outlives her husband?
A. 1/2
B. 4/7
C. 3/5
D. 2/3
E. 7/9
4. Suppose two losses are independent and uniformly distributed on
0 ≤ x ≤ 5 and 0 ≤ y ≤ 2, respectively. Find the expected value for
the maximum loss.
A. 37/18
B. 71/30
C. 79/30
D. 53/18
E. 55/18
5. The number of cars involved in an accident on a certain section of
highway is a Poisson random variable with mean 1.7. When the
number of cars is n, the damage is uniformly distributed on [1, n2].
Find the mean damage for a random accident.
A. 0.85
B. 1.95
C. 2.80
D. 3.31
E. 3.39
6. Suppose the number of sedans X and number of other vehicles Y a
4−x−y
family insures has joint probability function
with 0 ≤ x,
20
0 ≤ y, x + y ≤ 3. Find the variance of X + Y .
A. 1
B. 21/20
C. 22/15
D. 9/5
E. 33/10
4x + 6y
on 0 ≤ x ≤ 3, 0 ≤ y ≤
7. Losses X and Y have joint density
27
1. Find the probability that Y ≥ 1/2 given that X = 2.
A. 7/12
B. 25/44
C. 19/33
D. 32/55
E. 29/48
8. Find the mean of Y if X and Y have joint density x2e−x(y+1) on
0 < x, y.
A. 1
B. 10/7
C. 3/2
D. 12/7
E. 2
15 2
x y on 0 < x,
8
0 < y, x + y < 2. Let U = Y 2 − X 2 and V = X 2 + Y 2. Find the
value of the joint density function for U and V at U = 7/25, V = 1.
9. Suppose X and Y have joint density f (x, y) =
A. 0.1340
B. 0.1406
C. 0.2679
D. 0.27
E. 0.54
10. The number of claims handled by an agent in a day is a Poisson
random variable with mean 5. Each claim has probability 1/10 of
being reviewed by the agent’s supervisor. What is the probability
the supervisor reviews 3 such claims from a random day?
A. 0.0018
B. 0.0086
C. 0.0108
D. 0.0126
E. 0.0178
11. A claim is processed sequentially in two successive stages. The time
for the first stage is exponentially distributed with mean 1/2 day.
The time for the second stage is exponentially distributed with mean
1/3 day, independent of the time for the first stage. Find the probability the total time is between 1/2 day and 1 day.
A. 0.2307
B. 0.2762
C. 0.2991
D. 0.3239
E. 0.3509
12. Two independent policies have probability 0.2 of a claim being filed,
in which case the loss has uniform distribution on [0, 4]. Each policy
has deductible 1, If at least one claim has been filed, find the expected
total payment under the two policies.
A. 0.45
B. 1.11
C. 1.25
D. 1.45
E. 2.25