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Probability And Mathematical Statistics (Exam 1/P) Question Paper

Probability And Mathematical Statistics (Exam 1/P)

Course:Actuarial Science (Insurance)

Exam Year:

Joint Exam 1/P Sample Exam 3
Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer
for restroom breaks; Do not look at your notes. If you believe a question is defective or poorly worded,
you must continue on just like during the real exam.
Video solutions are available for this exam at http://www.theinniteactuary.com/?page=exams&id=50
TIA 1/P Seminar p. 1 Sample Exam 3
1. The average height of adult Americans is 176 cm, with a standard deviation of 6 cm, for males, and 163
cm, with a standard deviation of 5 cm, for females. If heights of each group are normally distributed,
what is the probability that a randomly selected American male is taller than a randomly selected
American female?
A. 0:85 B. 0:88 C. 0:91 D. 0:93 E. 0:95
2. The number of chocolate chips in a jumbo chocolate chip cookie at the Blue Frog Bakery has a binomial
distribution with mean 5 and maximum possible value 8. If I buy two jumbo chocolate chip cookies,
what is the coecient of variation for the total number of chocolate chips in the cookies?
A. 0:19 B. 0:27 C. 0:32 D. 0:38 E. 0:50
3. If P[A] = 0:7 and P[Bc] = 0:4, what is the maximum possible value of P[AB]?
A. :1 B. :3 C. :4 D. :6 E. :7
4. The joint moment generating function for X1 and X2 is
MX1;X2(t1; t2) = e??t1+2t2??3t1t2+4t21
+5t22
:
Find the correlation coecient between X1 and X2.
A. ??0:56 B. ??0:34 C. ??0:15 D. 0:34 E. 0:56
5. Three random variables X1;X2 and X3 have density 3x2 for 0 < x < 1. If the variables are i.i.d., nd
the density of the median of the three variables.
A. 3x2 B. 3x5 ?? 3x8 C. 18x5 ?? 18x8 D. 9x5 ?? 9x8 E. 8x3 ?? 7x6
6. The moment generating functions for X1 and X2 are given by
4
4 ?? t2 for ??2 < t < 2. If X1 and X2
are independent, what is the variance of X1 + X2?
A. 1=4 B. 1=2 C. 1 D. 2 E. 4
7. If X is an exponential random variable with mean 2, and Y =
p
X, nd fY (2), where fY denotes the
density of Y .
A. 0:07 B. 0:14 C. 0:18 D. 0:27 E. 0:37
8. Suppose that X and Y are discrete random variables taking the values 1; 2; 3 or 4, and that the joint
probability distribution, for all possible combinations of X and Y , is proportional to y3 + x2. Find
P[Y = 3 j X = 2].
A. 0:06 B. 0:12 C. 0:17 D. 0:22 E. 0:27
TIA 1/P Seminar p. 2 Sample Exam 3
9. Suppose that X and Y are uniformly distributed on the diamond 0 < jxj + jyj < 1. Find P[Y > 1=4 j
X = 1=2].
A. 0:25 B. 0:28 C. 0:39 D. 0:50 E. 0:75
10. Bob is always early to his company's weekly 8 am meeting, arriving at a time uniformly distributed
between 7:55 and 8:00. Charlie is always late to the same meeting, arriving at a time uniformly
distributed between 8 and 8:10. If their arrival times are independent, what is the probability that
they arrive within 5 minutes of each other?
A. 0 B. 1=8 C. 2=8 D. 3=8 E. 4=8
11. The cost of damage C in a re has a density given by f(c) = 3 2003=(c + 200)4 for c > 0. If the
damage is insured with is a $500 deductible, what is the expected payment?
A. 8 B. 16 C. 34 D. 92 E. 100
12. Suppose that X is an exponential random variable with mean , where is uniformly distributed on
[0; 2]. Find E
??
X2

.
A. 2=3 B. 1 C. 4=3 D. 2 E. 8=3
13. In a small, liberal arts college, 40% of the students have taken calculus. Of those who have taken
calculus, 25% have not seen Star Wars. Moreover, given that someone has not seen Star Wars, the
probability that that student has taken calculus is 20%. Find the probability that a randomly selected
student who has not taken calculus has seen Star Wars.
A. 1=5 B. 1=4 C. 1=3 D. 1=2 E. 2=3
14. Suppose that X and Y are uniformly distributed over the set y=3 < x < 2y and 0 < y < 20. Find
Var[Y j X = 10].
A. 75=4 B. 100=3 C. 625=12 D. 75 E. 175
15. For t < 2, nd the moment generating function for the random variable X whose density is 4xe??2x for
x > 0.
A.
4
(2 ?? t)2 B.
1
(1 ?? 2t)2 C.
4
4 ?? t2 D.
1
1 ?? 4t2 E.
4
(4 ?? t)2
16. The probability that a driver will get into an accident within 2 years of obtaining a drivers license
is 65% for males and 45% for females. If 4 people are randomly selected, 2 male and 2 female, and
their driving records are independent, what is the probability that at most 2 of them will get into an
accident within 2 years of obtaining a license?
A. 0:38 B. 0:47 C. 0:58 D. 0:61 E. 0:76
TIA 1/P Seminar p. 3 Sample Exam 3
17. The probability that I arrive at work on time on a given day is 25%. Suppose that I go to work 250 days
in a year, and whether or not I arrive on time each day is independent. Using a normal approximation
with a continuity correction, what is the approximate probability that I arrive on time more than 55
dierent days?
A. 0:83 B. 0:85 C. 0:87 D. 0:88 E. 0:91
18. If P[A \ B0] = 0:2, P[A \ B] = 0:3 and P[A [ B] = 0:8, nd P[A0 \ B].
A. 0:2 B. 0:3 C. 0:4 D. 0:5 E. 0:6
19. In Saint Tropez, the probability that it rains on a given day is 10%. Given that it rains, the amount
of rain has a density of f(x) = (1=3)e??x=3. Find the variance of the amount of rain on a given day.
A. 0:9 B. 1:7 C. 3:0 D. 4:4 E. 9:0
20. An urn contains 4 red balls, and 4 blue balls. A ball is randomly drawn from the urn, and then replaced,
along with a second ball of the same color. The process is then repeated. What is the probability that
the rst three balls drawn are two red balls and one blue ball?
A. 1=4 B. 1=3 C. 1=2 D. 2=3 E. 3=4
21. If the joint density of X and Y is proportional to x+y2 for 0 < x < 1 and 0 < y < 1, nd the variance
of Y .
A. 2=25 B. 7=60 C. 7=50 D. 33=80 E. 1=2
22. An urn contains 6 balls, 1 red, 2 blue, and 3 green. If I draw 2 balls with replacement, what is the
probability that they are dierent colors?
A. 8=36 B. 14=36 C. 18=36 D. 22=36 E. 28=36
23. The time T, in years, until the next time a piece of space debris with mass of at least 5g hits the Earth
has a probability density function fT (t) = 2e??2t. Let N be the number of years, rounded down, until
this occurs. What is VarN?
A. 0:14 B. 0:18 C. 0:25 D. 1:16 E. 1:34
24. An insurance company pays $10,000 for the rst loss, $7,500 for the second, and then $5,000 for each
successive loss. If the number of losses has a Poisson distribution with mean 2.5, what is the expected
loss size?
A. 12; 500 B. 14; 200 C. 15; 800 D. 17; 500 E. 18; 900
TIA 1/P Seminar p. 4 Sample Exam 3
25. Suppose that X and Y are jointly normal with E
??
X2

= 3;VarX = 2, E
??
Y 2

= 5, VarY = 4, and
that the correlation of X and Y is ??1=2. If the means of X and Y have dierent signs, nd EXY .
A. ??2:4 B. ??1:4 C. ??0:4 D. 0:6 E. 1:6
26. The cdf of X is given by
F(x) =
8>>>>>><
>>>>>>:
0 x 0
x
4 0 < x 2
2
3 2 x < 3
x=4 3 x < 4
1 4 x
Find EX.
A. 41=24 B. 47=24 C. 41=12 D. 43=12 E. 47=12
27. The joint density of X and Y is given by
fX;Y (x; y) =
5
81x2y
for 0 < x < y < 3, and is 0 otherwise. Find the marginal density function fY (y) for 0 < y < 3.
A.
5y4
243
B.
5y
9
C.
5x2
18
D.
5(9x2 ?? x4)
162
E.
5x4
162
28. If the joint density of X and Y is given by
fX;Y (x; y) =
(
xe??x(y+1) x > 0; y > 0
0 otherwise,
nd the conditional density of y given that X = 2.
A. 2e??2y B. xe??x(y+1)
e??x C. e??2 D. 2e??2y??2 E. 4ye??2y
29. The density f(x) of X is proportional to x2 for 0 < x < c, and is 0 otherwise. If the median of X is
2, what is c?
A. 2:0 B. 2:5 C. 3:0 D. 3:5 E. 4:0
30. Suppose that X is a Poisson random variable with mean 2. What is the probability that X is greater
than its mean, given that X is less than its mean plus its variance?
A. 4=19 B. 2=7 C. 5=7 D. 10=19 E. 12=19
TIA 1/P Seminar p. 5 Sample Exam 3
Answers
(1) E
(2) A
(3) D
(4) B
(5) C
(6) C
(7) D
(8) E
(9) A
(10) C
(11) A
(12) E
(13) C
(14) A
(15) A
(16) D
(17) B
(18) B
(19) B
(20) B
(21) A
(22) D
(23) B
(24) E
(25) A
(26) B
(27) A
(28) A
(29) B
(30) A
TIA 1/P Seminar p. 6 Sample Exam 3

Probability And Mathematical Statistics (Exam 1/P) Question Paper

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