April 25, 2016
MATH 260 HOMEWORK #7
(Due on Monday, May 2, 2016, submit your assignment to my office by 5:00 pm)
(Please, DO NOT forget to sign and hand in Declaration of Academic Honesty form)
1. A government agency is responsible for awarding funds to p
Fall 2016. Solutions of Homework 5 (Midterm 2 sample problems).
1. Let X be a random variable with the density function
1 2 x3
xe 9
3
if x > 0 and
0
otherwise
f (x) =
Find V ar(X 12 ).
Solution: X is a Weibull random variable with parameters = 3 and = 9.
Fall 2016. M250. Solutions of Homework 7.
(Sample Problems for Final Exam).
1. Let X be a normal random variable with EX = 10 and EX 2 = 116.
a) find the density function of Y = (X 10)2
b) find the density function of T = |X 10|.
Solution:
a) V arX = 116
M250. Section 2. Solution of QUIZ 8 (15.12.2016).
1. Let X be a gamma random variable with parameters = 2 and = 1 and given
X the random variable Y is uniformly distributed between X and 4X.
a) Find the density function fY (y).
b) Find P (X > 2|Y = 4).
So
Fall 2016. M250. Homework 7.
(Sample Problems for Final Exam, will not to be graded).
1. Let X be a normal random variable with EX = 10 and EX 2 = 116.
a) find the density function of Y = (X 10)2
b) find the density function of T = |X 10|.
2. Let X and Y
Solution of QUIZ 1. M250 Section 2. (10.10.2016).
1. How many ways are there to distribute 1 red, 1 white and 20 identical blue
balls into 10 distinct boxes such no box is empty?
Solution:
Case 1: red and white balls go to distinct boxes.
Choose a box for
Fall 2016. M250 Homework 3 ( Sample problems for Midterm 1) Solutions.
1. How many ways are there to distribute 1 red, 9 identical white and 8 identical
black balls into 20 distinct boxes so that exactly three boxes are empty?
Solution: Since exactly 3 bo
Solution of QUIZ 3. M250 Section 2. (24.10.2016).
1. A box contains 8 balls numbered 1, 2, . . . , 8. We randomly remove one ball.
Let X and Y be the smallest and greatest numbers in the box after removal. Find
V ar(Y X).
Solution: R(Y X) = cfw_6, 7. fY X
M250. Section 2. Solution of QUIZ 7 (28.11.2016).
1. Let X be a random variable with the following density function:
f (x) =
1 x2
xe 10
5
if x > 0
0
otherwise
Find VarX 4 .
Solution: X is a Weibull random variable with parameters = 2 and = 10.
V arX 4 = E
Fall 2016 M250 Homework 2 Solutions.
1. Suppose we have four chests each having two drawers. Chests 1 and 2 have a
gold coin in one drawer and a silver coin in the other drawer. Chest 3 has two gold coins
and chest 4 has two silver coins. A chest is selec
Fall 2016. M250. Solutions of Quizzes, 1,2,3. Sections 1,2,3.
1. How many ways are there to distribute 2 identical red and 20 identical white
balls into 10 distinct boxes such that at least one box is empty?
Solution:
We can distribute 2 red and 20 white
Fall 2016. M250 Homework 6. Due to Thursday, December 29, 2016.
TIME: 15.30.
1. The random variables X and Y have a joint density function given by:
f (x, y) =
Cxy if 0 x 1, 0 y 1 and
0
otherwise
a) Determine C and find P (X > Y > 12 ).
b) Let U = max(X1
Fall 2016. M250 Homework 2. Due to Monday, October 31, 2016,
TIME: 16.30.
1. Suppose we have four chests each having two drawers. Chests 1 and 2
have a gold coin in one drawer and a silver coin in the other drawer. Chest 3 has
two gold coins and chest 4 h
Fall 2015. M250 Homework 3: Sample problems for Midterm 1.
(will not be graded, solutions will be sent soon ).
1. How many ways are there to distribute 1 red, 9 identical white and 8 identical
black balls into 20 distinct boxes so that exactly three boxes
Fall 2016. M250 Homework 5: Sample problems for Midterm 2.
(will not be graded, solutions will be sent soon).
1. Let X be a random variable with the density function
f (x) =
1 2 x3
xe 9
3
if x > 0 and
0
otherwise
Find V ar(X 12 ).
2. a) Let X be uniformly
2014 Fall. Solutions of M250 Midterm 2 problems.
1.
Let X and Y be independent random variables having the following
density functions:
C1 sinx if 2 x
0
otherwise
C2 siny if 0 y
0
otherwise
f (x) =
f (y) =
a) Determine C1 and C2 .
b) Find the distribu
M250 Section 1. Solution of QUIZ 5 (13.11.2014).
1. Let X be gamma a random variable with EX = 24 and V arX = 192. Find EX 3 .
Solution: Let X be gamma random variable with parameters and . Since EX =
k
, we get
and V arX = 2 we get = 3 and = 8. Now sinc
October 9, 2015
Bilkent University
Department of Industrial Engineering
IE 262 Self Study Homework Questions- Set 2
Answers will be posted on October 13th, 2015
1. You are given the K and n values of two different materials. Is this information sufficient
2016 Fall. Solutions of M250 Final Exam problems.
1. The joint density function of X and Y is given by
3
if 0 x 2, x2 y 4
16
0
f (x, y) =
elsewhere
a) Find the density function of T = max(X, Y ).
Solution: R(T ) = [0, 4].
F (t) = P (X t, Y t) =
Z tZ t
0
Solution of QUIZ 2. M250 Section 2. (17.10.2016).
1. We toss a fair coin three times. Given that the face H has appeared at
least once find the probability that the face T has appeared exactly once.
Solution:
Let A be the event that the face H has appeare
M250 Section 2. Solution of QUIZ 6 (21.11.2014).
1. Let X be a random variable with the following density function:
1 x2
xe 6
3
if x > 0 and
0
otherwise
f (x) =
Find V arX 4 .
Solution: X is a Weibull random variable with parameters = 2 and = 6.
V arX 4
Fall 2016. M250 Homework 1. Solutions.
1. How many ways are there to partition 9 distinct objects into 3 indistinguishable
boxes so that at least one box contains 3 objects?
Solution: Possibilities: (3,3,3), (3,6,0), (3,5,1), (3,4,2).
!
!
!
!
!
9!
9
9
6
9
M250. Section 2. Solution of QUIZ 5 (14.11.2016).
1. Let X be exponential random variable and P (X > 12) = e3 . Find
V ar(X 3 + 2016).
Solution:
12
P (X > 12) = e = e3 . Therefore, = 4.
V ar(X 3 + 2016) = V arX 3 = EX 6 (EX 3 )2 = 6!6 (63 )2 = 6846 =
2801
M250. Section 2. Solution of QUIZ 4 (10.11.2016).
1. Let X be a random variable with the following density function:
(
f (x) =
a + bx if 0 x 1
0
otherwise
Find P (X > 49 ) if EX = 32 .
Solution:
EX =
R1
0
R
f (x)dx =
R1
0
(a + bx)dx = 1. Therefore, a +
a
Fall 2016. M250 Homework 1. Due to Monday, December 5, 2016,
TIME: 16.30.
1. Let X be uniformly distributed over [5, 1] and Y be the distance from
X to the nearest endpoint of [5, 1]. Find the variance of T = Y 2 + Y + 2016.
2. Let X and Y be independent