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 timerfor 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.theinfiniteactuary.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 163cm, 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 selectedAmerican 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 binomialdistribution with mean 5 and maximum possible value 8. If I buy two jumbo chocolate chip cookies,what is the coefficient 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 coefficient 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., findthe 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 by4

4− t2for −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 =√X, find 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 jointprobability distribution, for all possible combinations of X and Y , is proportional to y3 + x2. FindP[Y = 3 | 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 < |x|+ |y| < 1. Find P[Y > 1/4 |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 distributedbetween 7:55 and 8:00. Charlie is always late to the same meeting, arriving at a time uniformlydistributed between 8 and 8:10. If their arrival times are independent, what is the probability thatthey 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 fire has a density given by f(c) = 3 · 2003/(c + 200)4 for c > 0. If thedamage 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 θ, where θ 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 takencalculus, 25% have not seen Star Wars. Moreover, given that someone has not seen Star Wars, theprobability that that student has taken calculus is 20%. Find the probability that a randomly selectedstudent 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. FindVar[Y | X = 10].

A. 75/4 B. 100/3 C. 625/12 D. 75 E. 175

15. For t < 2, find the moment generating function for the random variable X whose density is 4xe−2x forx > 0.

A.4

(2− t)2B.

1(1− 2t)2

C.4

4− t2D.

11− 4t2

E.4

(4− t)2

16. The probability that a driver will get into an accident within 2 years of obtaining a drivers licenseis 65% for males and 45% for females. If 4 people are randomly selected, 2 male and 2 female, andtheir driving records are independent, what is the probability that at most 2 of them will get into anaccident 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 daysin a year, and whether or not I arrive on time each day is independent. Using a normal approximationwith a continuity correction, what is the approximate probability that I arrive on time more than 55different days?

A. 0.83 B. 0.85 C. 0.87 D. 0.88 E. 0.91

18. If P[A ∩B′] = 0.2, P[A ∩B] = 0.3 and P[A ∪B] = 0.8, find P[A′ ∩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 amountof 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 thatthe first 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, find the varianceof 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 theprobability that they are different 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 Earthhas a probability density function fT (t) = 2e−2t. Let N be the number of years, rounded down, untilthis 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 first loss, $7,500 for the second, and then $5,000 for eachsuccessive loss. If the number of losses has a Poisson distribution with mean 2.5, what is the expectedloss 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, andthat the correlation of X and Y is −1/2. If the means of X and Y have different signs, find 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) =

0 x ≤ 0x4 0 < x ≤ 223 2 ≤ x < 3x/4 3 ≤ x < 41 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) =581x2y

for 0 < x < y < 3, and is 0 otherwise. Find the marginal density function fY (y) for 0 < y < 3.

A.5y4

243B.

5y9

C.5x2

18D.

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 > 00 otherwise,

find the conditional density of y given that X = 2.

A. 2e−2y B.xe−x(y+1)

e−xC. 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 is2, 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 greaterthan 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