ORIE 3500/5500 Engineering Probability and Statistics II
Fall 2012
Assignment 12
Problem 1 Suppose that the number of accidents occurring daily
in a certain plant has a Poisson distribution with an unknown mean
. Based on previous experience in similar in
OR 3500/5500, Fall12, Assignment 12 Solutions
Assignment 12 Solutions
Problem 1 Solution: Let X Poiss() be the number of accidents
xi
occurring daily in the plant. Then pX (xi ; ) = e xi for i = 1, . . . , n,
!
and for > 0,
n
p()
pX (xi ; ) =
i=1
2e2 en
OR 3500/5500, Fall12, Assignment 11 Solutions
Assignment 11 Solutions
1. We have the following likelihood function:
n
n
L(x) =
1 xi 
e
=
2
i=1
fX (xi ) =
i=1
l(x) = n log
1
2
1
2
n
e
n
i=1
xi 
n
xi 
i=1
n
So we have to minimize the function i=1 x
OR 3500/5500, Fall14, Homework 9
Homework 9
Problem 1. Let us assume that the prior distribution for the proportion of
drinks from a vending machine that overflow is
0.05
p() 0.3
0.10
0.5
0.15
0.2
If 2 of the next 9 drinks from this machine overflow, find
ORIE 3500/5500 Engineering Probability and Statistics II
Fall 2012
Assignment 11
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
OR 3500/5500, Summer 15, Homework 5 Solutions
Homework 5
Solutions
Problem 1.
a) The Exponential random variable with parameter > 0 is a continuous
random variable with density
cfw_
ex if x 0,
fX (x) =
0
if x < 0.
We denote X Exp(). Compute EX.
b) A media
OR 3500/5500, Summer 15, Homework 6
Homework 6
Solutions
Problem 1. Let X and Y be two random variables jointly distributed ac2
cording to a bivariate normal distribution with parameters X , Y , X
, Y2 and
as shown on slide 14 of Lecture 6. Find the mome
OR 3500/5500, Summer 15, Homework 9, Not to Hand In
Homework 9
Not to Hand In
Solutions
Problem 1.
Suppose that the time to failure T of a certain hinge is an
exponential random variable with mean 1/. From prior experience we are led
to believe that is a
OR 3500/5500, Fall12, Prelim 1 Solutions
Prelim 1 Solutions
PROBLEM 1
Denote by H the event that the coin comes up heads, and T otherwise.
Denote by W the event that at least one white chip was drawn.
Then P (H ) = 2/3 and P (T ) = 1/3. Also,
P (W H ) =
ORIE 3500/5500 Engineering Probability and Statistics II
Fall 2012 Practice problems for Midterm 1
Problem 1 Two identical boxes contain (crooked) coins. Each coin
in box 1 shows heads with probability 1/4, and tails with probability
3/4, while each coin
ORIE 3500/5500 Engineering Probability and Statistics II
Fall 2012 Practice problems for Midterm 2
Problem 1 Suppose that the daily maximal air temperatures X1
and X2 , observed on two successive days follow a bivariate normal
distribution with parameters
ORIE 3500/5500 Engineering Probability and Statistics II
Fall 2012 Practice problems for Final Exam
Problem 1 Let X1 , . . . , Xn be a sample from the uniform distribution
on the interval (0, a), where a is an unknown parameter. We put the
uniform in the
Prelim 2 Solutions
1. First, the density of X is
fX (x) =
1
, for 1 < x < 1.
2
and 0 otherwise.
For 1 < x 0, we have the following transformation:
T1 (x) = x3
The inverse transformation is for 1 < y 0,
T1 1 (y ) = y 1/3
We have for y = 0,

1
1
d 1
T1 (y
OR 3500/5500, Fall'12, Assignment 1 Solutions
Assignment 1 Solutions
1. P (A) = 0.7, P (B) = 0.5, P (A B) = 0.3 P (Ac B) P (A B ) P (A B )
c c c
= = =
P (B\A) = 0.5  0.3 = 0.2 P (A\B) = 0.7  0.3 = 0.4 P (A B)c ) = 1  0.9 = 0.1
P (A B) P (Ac B) P (A B c
OR 3500/5500, Fall'12, Assignment 2 Solutions
Assignment 2 Solutions
1 1 1. First, we have P (A) = 10 = 2 , P (B) = 10 = 1 and P (C) = 10 = 2 . 20 20 2 20 However, P (A B C) = 0 = P (A)P (B)P (C), so A, B and C are not independent.
Let X be the number cho
OR 3500/5500, Fall12, Assignment 7 Solutions
Assignment 7 Solution
1. (a) X exp(1), Y = log(X )
y
= P (log(X ) y ) = P (X ey ) = FX (ey ) = 1 ee
y
y
y
d
d
fY (y ) =
1 ee = ee ey = eye , y R
P (Y y ) =
dy
dy
P (Y y )
(b) X = eY fY (y ) = fX (ey )
dy
dy e
y
OR 3500/5500, Summer 15, Homework 7
Homework 7
Solutions
Problem 1. Text: #26.3, p. 394
Solution:
a) The probability of a type I error in this case is the probability that a single
observation of a Uniform(0, ) random variable, X, falls outside the interv
OR 3500/5500, Summer 15, Homework 8
Homework 8
Solutions
Problem 1. Text: 19.1, p. 294.
Problem 2. Text: 19.2, p. 294.
Problem 3. Text: 20.5, p. 308.
Problem 4. Text: 20.9, p. 309.
Problem 5. Text: 21.5, p. 324.
Problem 6. Let X1 , , Xn denote a random sa
ORIE 3500/5500, Fall 2016, Homework 7
Homework 7 Due October 28th
Read Chapter 8 and Reread Chapter 11 of the textbook.
Problem 1. If X Gamma(n, 1), how large should n be so that
X
P 1 > 0.01 < 0.01
n
a) according to Chebyshevs inequality?
Problem 2.
Y?
ORIE 3500/5500, Fall 2016, Homework 4
Homework 4 Due September 30th
Problem 1.
Let X and Y be independent standard normal random variables, that is,
they both have probability density function given by
t2
1
fX (t) = fY (t) = e 2 , t R.
2
Let U = X + Y and
ORIE 3500/5500, Fall 2016, Homework 5
Homework 5 Due October 14th
Problem 1. Let X and Y be two independent random variables, where X
has a Bernoulli distribution with probability of success p = 1/2 and Y has a
Bernoulli distribution with probability of s
ORIE 3500/5500, Fall 2016, Homework 2
Homework 2 Due September 16th
Reading :
Read Chapters 3 and 4 of the textbook.
Problem 1. Suppose that the waiting time (in minutes) to be served at a
counter in a bank is a continuous random variable X having the den
ORIE 3500/5500, Fall 2016, Homework 1
Homework 1 Due September 2nd
Reading :
What Are The Odds? (Article on Blackboard)
Read Chapters 1 and 2 of the textbook.
Problem 1. Five cards are selected from a 52card deck for a poker hand.
Find the probability of
ORIE 3500/5500, Fall 2016, Homework 6
Homework 6 Due October 21st
Read Chapter 11 of the textbook.
Problem 1. Exercise 11.1 in textbook.
Problem 2. Exercise 11.2 in textbook.
Problem 3. A continuous random variable X has a Pareto distribution with
paramet
ORIE 3500/5500, Fall 2016, Homework 1
Homework 1
Solutions
Problem 1. Five cards are selected from a 52card deck for a poker hand.
Find the probability of the following poker hands.
a) Four of a kind. This is, four cards of the same denomination and any
ORIE 3500/5500, Fall 2016, Homework 2
Homework 2
Solutions
Problem 1. Suppose that the waiting time (in minutes) to be served at a
counter in a bank is a continuous random variable X having the density function
if x < 0,
0
fX (x) = 1/2
if 0 x < 1,
3/(2x4
ORIE 3500/5500, Fall 2016, Homework 3
Homework 3 Due September 23rd
Problem 1. Let pi = P (X = i). and
a) Suppose that p1 + p2 + p3 = 1. If EX = 2, what values of p1 , p2 and p3 (i)
maximize and (ii) minimize Var(X)?
b) Suppose now that p1 + p2 + p3 + p4
ORIE 3500/5500, Fall 2016, Homework 8
Homework 8: Due November 5th at 5pm
Problem 1. Let X and Y be the coordinates of a point selected at random
inside the triangle
T = cfw_(x, y)  y 0, y 1 + x, y 1 x.
That is, the joint pdf of X and Y is
(
fX,Y (X, Y )
ORIE 3500/5500, Fall 2016, Homework 5
Homework 5
Solutions
Problem 1. Let X and Y be two independent random variables, where X
has a Bernoulli distribution with probability of success p = 1/2 and Y has a
Bernoulli distribution with probability of success