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 bet in the following manner. If the rst head occurs on th
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 2 y dx dy = , 3 9
EY 2 =
0
7 2 , y 2 dx dy = 3 18 1 13
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 ; T T T (b) Ac = cfw_HHH; T HH; HT H; HHT ; T T T A (C
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 chosen in the set of k coupons. Then, P (A) = (1  pi ) ,
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 table are equal to 14 . Since P (X = 1) = 14 , we know 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
1 1 y) = P (X y) + P (X ) X y
1
= 1  ey + (1  (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) = 1/3 . Now, we have E(X) = E(E(XY ) = E(XY = 0)P (
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 distribution whose density function is 1 fX (x) = ex
ORIE 3500/5500, Fall '10
HW 12 Solution
Assignment 12 Solution
Problem 1 The joint pdf of X1 , . . . , Xn is given by
n
fX1 ,.,Xn (x1 , . . . , xn ; ) =
i=1
fXi (xi ; ) =
1 1 xi  e = n e 2 2 i=1
n
n i=1
xi 
.
Taking the logarithm of this expressio
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 variable with parameters n and p, then pX (k) = k1 n p
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 + (yY )  2 (xX Y Y ) 2 2 1 X Y . fX,Y (x, y) = exp 
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. Show your work. GOOD LUCK!
Problem 1 Two identical boxe
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 from different boxes; HH: both coins show tail; T T : bo
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 B comes up heads. Let SA and SB denote the events that
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 by:
P= C C + F =C + Fn n t n (1 + r ) (1 + r ) r r (1 +
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 . Let Y denote the amount of money won on a side bet in
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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 . otherwise
(a) Find the mean, variance and standard deviation
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 the maximum likelihood estimator of . (b) Compute the
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 2 + y) 10
0 y=2 y=0
1 (4x + 2y + 1) dy 10 1 (8x + 6) 1
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=1
xi
.
So the loglikelihood function is
n
log L() = n
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 coins look the same). You toss each of the two coins t
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. Show your work. GOOD LUCK!
Problem 1 A continuous rando