ORIE 3500/5500, Fall '10
Prelim 2 Solution
Prelim 2 Solution
Problem 1 (a)
1 y 1 1 y
EX =
0
2 2 x dx dy =  , 3 9
EX 2 =
0
2 5 x2 dx dy = , 3 18 1 37 , 162
VarX = EX 2  (EX)2 =
1 y 1 y
EY =
0 1
5
ORIE 3500/5500 Engineering Probability and Statistics II Fall 2010 Assignment 12 Problem 1 Find the maximum likelihood estimator of the unknown parameter when X1 , X2 , . . . , Xn is a sample from the
ORIE 3500/5500, Fall '10
HW 1 Solutions
Assignment 1 Solutions
Problem 1 (a) A = cfw_T T H; T HT ; HT T B = cfw_T T H; T HT ; HT T ; T T T C = cfw_HHH; HT H; HHT ; HT T D = cfw_T HH; T T H; T HT ;
ORIE 3500/5500, Fall '10
HW 2 Solutions
Assignment 2 Solutions
Problem 1 Let A be the event that a type i coupon is not chosen in the set of k coupons and B be the event that a type j coupon is not ch
ORIE 3500/5500, Fall '10
HW 3 Solutions
Homework 3 Solutions
Problem 1 1 Since P (X = ai , Y = bj ) is either 0 or 14 , we know that all the entries of the first 1 1 column and the second row of the t
ORIE 3500/5500, Fall '10
HW 9 Solutions
Assignment 9 Solutions
Problem 1 Clearly, Y takes values in [0,1]. So FY (y) = 0 for y 0 and FY (y) = 1 for y 1. Now for y (0, 1) FY (y) = P (Y y) = P (X y or
1
ORIE 3500/5500, Fall '10
HW 10 Solutions
Assignment 10 Solutions
Problem 1 Let X be the time in jail and Y be the result of the first draw. From the information given, P (Y = 0) = P (Y = 1) = P (Y = 3
Useful formulas
If X is a binomial random variable with parameters n and p, then pX (k) = n k p (1  p)nk , k k = 0, 1, . . . , n.
E(X) = np , Var(X) = np(1  p). If X is a negative binomial random
OR 3500/5500, Fall'10
Formula sheet
Formulas related to Bivariate Normal Distribution
If (X, Y ) is bivariate normal with parameters (X , Y , X , Y , ), then its density is given by 2 (xX )2 X )(y
ORIE 3500/5500 Engineering Probability and Statistics II Fall 2010 Midterm 1 The question sheet has two sides. The exam is closed books and notes. You have 90 minutes. All problems have equal weight.
ORIE 3500/5500 Engineering Probability and Statistics II Fall 2010 Midterm 2 The question sheet has two sides. The exam is closed books and notes. You have 90 minutes. All problems have equal weight.
ORIE 3500/5500, Fall '10
Prelim 1 Solution
Prelim 1 Solution
Problem 1 Introduce events B11 : the two coins were taken from box 1; B22 : the two coins were taken from box 2; B12 : the two coins came f
ORIE 3500/5500, Fall '10
Solutions for Practice Problems
Solutions for Selected Practice Problems
Problem 1 Let XA be the number of times coin A comes up heads, and let XB be the number of times coin
01/07/2008 Christos Papahristodoulou, Mlardalen University/HST/Economics
01/07/2008
Duration, convexity and portfolio immunization
Some principles of bonds prices As is known, a bonds price is given b
University of California, Los Angeles Department of Statistics Statistics 100A Homework 6
EXERCISE 1 A coin is tossed 3 times independently. One of the variables of interest is the number of tails X .
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Practice problems for Midterm 2 Problem 1 The amount X in pounds of polyurethane cushioning found in a car is modeled as a continuous random variable with pdf fX (x) = 0
1 1 ln 2 x
if 25 x 50 . otherw
Practice problems for Final Exam Problem 1 A sample X1 , X2 , . . . , Xn comes from a continuous distribution with the density fX (x) = ( + 1)x , 0 < x < 1 for some unknown parameter > 1. (a) Compute
ORIE 3500/5500, Fall '10
Solutions for Practice Problems
Solutions for Selected Practice Problems
Problem 2 (a) The marginal densities of X and Y are given by
2
fX (x) =

fX,Y (x, y) dy = 1 (4xy + y
ORIE 3500/5500, Fall '10
Solutions for Practice Problems
Solutions for Selected Practice Problems
Problem 1 (a) The likelihood function L() for > 1 is given by
n n
L() =
i=1
fX (xi ; ) = ( + 1)
n i=
Practice problems for Midterm 1 Problem 1 You are given two coins, one fair and the second one that comes up heads with probability .7. Initially, you have no information which coin is biased (the two
Statistics 265 Assignment 5 Solutions
5.2 Three balanced coins are tossed independently. One of the variables of interest is Y1; the number of heads. Let Y2 denote the amount of money won on a side be