ORIE 360/560 { Engineering Probability and Statistics II Fall 2003 FinalExam
All problems have equal weight. SHOW YOUR WORK. GOOD LUCK!
ables, X with parameter 1, and Y with parameter 2. Let V = X + Y and W = X=Y . (a) Find the joint density of V an

Homework 1
September 27, 2007
Problem 1.
1. = {a1 a2 a3 a4 a5 : aj {H, T }, for j = 1, . . . , 5}. | = 32. 2. P { the rst 3 ips are the same} = 2 0.53 = 0.25 3. Let A be the event that the rst three ips are the same; let B be the event that the

ORIE 360/560 Fall 2007 Assignment 2
Due Tuesday, Sept. 11 at 2:00 pm
Problem 1. A traveler puts her passport in one of n drawers in her desk, but has forgotten which one. Before starting a journey, she is frantically looking for it. She tries the dra

ORIE 360/560 Fall 2007 Assignment 3
Due Tuesday, Sept. 18 at 2:00 pm
Problem 1. In a certain engineering class every student receives a grade of 1.0, 2.0, 3.0, or 4.0., A student has a 30% probability of receiving a failing grade of 1.0 and a 70% pro

Homework 3
September 17, 2007
Problem 1.
Let A =A student passes the course P (A) = 0.7 and P (x = 1.0) = P (AC ) = 0.3,
Thus X is discrete.
P (x = 2.0) = P (x = 2.0 A) = P (x = 2.0|A) P (A) = 0.7 0.5 = 0.35 P (x = 3.0) = P (x = 3.0 A) = P (

ORIE 360/560 Fall 2007 Assignment 4
Due Tuesday, Sept. 25 at 2:00 pm
Please remember to show your work! Please remember to put your section number and net id on the assignment! Problem 1. Suppose X has pdf f (x) = a + bx2 , for 0 x 1. Let c = EX. F

ORIE 360/560 { Engineering Probability and Statistics II Fall 2003 Midterm 2
All problems have equal weight. SHOW YOUR WORK. GOOD LUCK!
(4x + 2y + 1) if 0 < x < 1 and 0 < y < 2 : 0 otherwise (a) Compute Cov(X; Y ) and the correlation . (b) Find the

ORIE 360/560 Engineering Probability and Statistics II Fall 2005 Midterm 2 All problems have equal weight. SHOW YOUR WORK. You have 90 minutes. GOOD LUCK! Problem 1 In baseball, the World Series lasts until one team wins 4 games. This year Chicago W

OR 3500/5500, Summer'08
Practice Problems
Not to be turned in. Problem 1 You have a fair coin and you keep tossing it until you get 5 heads. (a) Find the probability that you will need to toss the coin exactly 20 times. (b) If X is the random variab

OR 3500/5500, Summer'08, Homework 1
Homework 1
Due on Tuesday, May 27, 11am. For each problem just giving the answer will not suffice; a proper argument is required. Problem 1 A fair die is rolled twice. (a) List the elements in the following events

Homework 2
Due on Friday, May 30, 11am. For each problem just giving the answer will not suffice; a proper argument is required. Problem 1 A lock on a lab door has buttons numbered 0 through 9. The right code for opening the lock is 8542. Also, if on

OR 3500/5500, Summer'08, Homework 3
Homework 3
Due on Monday, June 2, 4pm. Problem 1 Suppose that X is a continuous random variable with the following density f (x) = 1/3, -2 x 1 Compute the density of X 2 . Solution We first compute the cdf of X

ORIE 360/560 Practice Final
Fall, 2007
Problem 1. Suppose that X is uniform on (0, Y ), where Y itself is random and equals 1 or 2 each with probability 1/2. (a) Find EX and Var X. (b) Find P (X > 1). Problem 2. A bridge hand consists of 13 randomly

ORIE 360/560 Fall 2007 Assignment 10
Due Tuesday, Nov. 27 at 2:00 pm
Problem 1. Consider a normal (, 2 ) population distribution with the value of 2 known. (a) What is the condence level for the interval x 2.81/ n? (b) What is the condence level

ORIE 360/560 { Engineering Probability and Statistics II Fall 2003 Midterm 1
All problems have equal weight. SHOW YOUR WORK. GOOD LUCK! Problem 1 A biased coin, twice as likely to come up heads as tails, is tossed once. If it shows heads, two chips a

ORIE 360/560 Engineering Probability and Statistics II Solution of Fall 2006 Prelim 1 Problem 1 Let A = {the chosen die is crooked}, B = {the chosen die shows a 4 two times in a row}. We are looking for the conditional probability P (A|B). By the Ba

Homework 4
September 21, 2007
Problem 1.
Notice that
1 0 f (x)dx 1
= 1 and
1 0 xf (x)dx 1
= c. Then
(a + bx2 )dx = 1,
0 0
x(a + bx2 )dx = c.
Arranging these equations, we have
a+ b = 1, 3 ab + = c. 24
Hence, a = 4c + 3 and b = 12c 6.
Pro

ORIE 360/560 Fall 2007 Assignment 5
Due Friday, Oct. 5 at 2:00 pm
Please remember to show your work! Please remember to put your section number and net id on the assignment! Problem 1. This problem extends the investment example from class. Suppose y

Homework 5
October 10, 2007
Problem 1.
Denote the variance as g(a).
g(a) Var(aX + (20000 a)Y )
2 2 = a2 X + (20000 a)2 Y + 2a(20000 a)X Y
dg(a) 2 2 = 2aX 2(20000 a)Y + 2(20000 a)X Y 2aX Y = 0 da
Solving this equation, we have
a=
2 20000(

ORIE 360/560 Fall 2007 Assignment 6
Due Tuesday, Oct. 16 at 2:00 pm
Please remember to show your work! Please remember to put your section number and net id on the assignment! Problem 1. Suppose X has an exponential distribution, and that P [X > .01]

Homework 6
October 17, 2007
Problem 1.
Let be a rate parameter. Then notice that P (X > .01) = 1 P (X .01) = e0.01 . Solving 1/2 = e0.01 , we have = 100 ln 2. Thus, P (X > t) = e100t ln 2 , so by solving .9 = e100t ln 2 , we have
t = 0.01 log2

ORIE 360/560 Fall 2007 Assignment 7
Due Tuesday, Oct. 23 at 2:00 pm
Please remember to show your work! Please remember to put your section number and net id on the assignment! Problem 1. The number of minutes airline ights are behind schedule is assu

ORIE 360/560 Fall 2007 Assignment 8
Due Tuesday, Oct. 30 at 2:00 pm
Problem 1. (a) Suppose (U, V ) is uniformly distributed on the triangle with vertices (0, 0), (1, 0), (1, 1). Find E[V |U ] and E[V ]. (b) Suppose X is uniform on (0, 1) and, conditi

ORIE 360/560 Fall 2007 Assignment 9
Due Thursday, Nov. 15 at 2:00 pm
Problem 1. Consider a casino in which gamblers are playing on identical slot machines. Assume each machine produces a jackpot on a particular pull of the level, independently from o

Homework 9
December 7, 2007
Problem 1.
Let Xi be the number of pulls. Xi follows a geometric distribution with parameter p. Thus the likelihood function is f (p) = (1 p) X 20 p20 (i = 1, 2, , 20 )
i
Set In this case,
Problem 2.
log f (p) = (