MA 2621: Probability for Applications
Homework 2
Due on January 29 in class
Please staple your work before submission.
Total: 100 points
1. A certain system can experience three different types of defects. Let Ai (i = 1, 2, 3) denote the event
that the sy

Exercise 7.9
Solutions to exercises - Week 40
X 1 ,., X n are iid with pdf
f ( x | ) = 1/
Moment and maximum likelihood estimators:
for 0 x
and VarX = 2 /12
Exercises 7.9, 7.11 and 7.13
We have that EX = / 2
Bayes estimators:
The moment estimator is gi

MA4632/541
Spring 2017
HOMEWORK ASSIGNMENTS
Homework #2
Assigned: 1/19/17
Due: 2/9/17
[Under construction!]
1. You tossed two six-sided dice and constructed a statistic that is the sum of the
two faces that occurred. Explain how this statistic partitions

Introduction to Probability
Example Sheet 2 - Michaelmas 2006
Michael Tehranchi
Problem 1. Let X1 , X2 , . . . be independent and identically distributed exponential random
variables with parameter . Prove
lim sup
n
1
Xn
=
log n
almost surely.
Solution 1.

Stat 821
Homework 3
Solution
Question 1
(a)
n
[x1
I(0,1) (xi )]
i
f (x) =
i=1
(
= n
n
)1
xi
i=1
n
I(0,1) (xi )
i=1
)1
[
] (
n
= I(0,1) (mincfw_x1 , . . . , xn )I(0,1) (maxcfw_x1 , . . . , xn )
xi
n
i=1
By Neymans factorization theorem, T =
n
xi is the

MA4632/541
Spring 2017
HOMEWORK ASSIGNMENTS
Homework #3
Assigned: 2/9/17
Due: 3/2/17
[Under construction!]
iid
1. Let X1 , . . . , Xn | 2 Normal(0, 2 ). Let T =
complete.
Qn
i=1
Xi . Show that T is not
2. Exercise 6.26 in the Casella and Berger textbook.

Homework 4
MA 2621: Probability for Applications
Due on Feb. 5 in class or office hours 2-4pm
Please staple your work before submission.
1. Consider the experiment of rolling two dice. Let random variable M be the product of the two numbers
obtained.
(a)

Homework 5
MA 2621: Probability for Applications
Due on Feb. 12 in class or office hours 2-4pm
Please staple your work before submission. Indicate your section number.
Total: 55 points
1. If E(X) = 1 and Var(X) = 5, find
(a) Var(2 + 3X). (5 pts.)
Ans:
Var

MA 2621: Probability for Applications
Due on Feb. 16 in class or office hours 3-5pm
Homework 6
Please staple your work before submission.
2. Suppose we draw 13 cards from a deck of 52 cards.
How many outcomes lead to a 6-3-2-2 distribution, that is, a han

MA2621: Probability for Applications
Midterm II
Name:
Total : 70 points.
1. Suppose the defective rate of products from a supplier is 1%. Suppose the quality of products are
independent. Each time we sample one product and check if it is defective or not.

MA 2621: Probability for Applications
Homework 2
Due on January 29 in class
Please staple your work before submission.
Total: 100 points
1. A certain system can experience three different types of defects. Let Ai (i = 1, 2, 3) denote the event
that the sy

Homework 8
MA 2621: Probability for Applications
Due on Feb. 26 in class or office hours 2-4pm
Please staple your work before submission. Indicate your section number.
Total: 60 points.
1. Binary digits, that is, 0s and 1s, are sent down a noisy communica

MA 2621: Probability for Applications
Homework 3
Due on Feb. 2 in class
Please staple your work before submission. Indicate your section number.
1. Each time a component is tested, the trial is a success (S) or failure (F ). Suppose the component is
teste

MA 2621
PROBABILITY FOR APPLICATIONS
TERM A -2016
INSTRUCTOR: Buddika Peiris, PhD (e-mail : [email protected] )
LECTURE: MTRF 2.00 -2.50 pm, OH 107
OFFICE: SH 100 (phone: 508 831 5940)
OFFICE HOURS: MTRF 10.00-10.50 am (or by appointment)
CONFERENCE:
TA: X

Homework 4
MA 2621: Probability for Applications
Due on Feb. 5 in class or office hours 2-4pm
Please staple your work before submission.
Total: 70 pts.
1. Consider the experiment of rolling two dice. Let random variable M be the product of the two numbers

Homework 9
MA 2621: Probability for Applications
Due on March. 1 in class or office hours 3-5pm
Please staple your work before submission. Indicate your section number.
Total: 55 points
1. Time headway in traffic flow is the elapsed time between the time

Homework 5
MA 2621: Probability for Applications
Due on Feb. 16 in class or office hours 3-5pm
Please staple your work before submission.
Total: 100 points
1. Four red, 8 blue, and 5 green balls are randomly arranged in a line. Assume all the balls are di

Homework 7
MA 2621: Probability for Applications
Due on Feb. 19 in class or office hours 2-4pm
Please staple your work before submission.
1. Consider a simple gambling game. Each you bet $10. If you lose (with probability 1 p), you lose the
$10 you bet; i

MA4632/MA541
Spring 2017
Probability and Mathematical Statistics II
COURSE OUTLINE
1/12/17
Instructor: Balgobin Nandram; Voice: 831-5539, F: 831-5824; Email: [email protected];
Web: http:/www.wpi.edu/ balnan
Office: SH 002A; Office Hours: Thu 2-4; other time

Theorem A standard uniform random variable X can be transformed to a log logistic
random variable Y through the transformation
Y =
1 1X
X
1/
,
where and are positive.
Proof Let the random variable X have the standard uniform distribution with probability

Peary-
e, -.-_ (rms) 0 (Anm
=5 pun = puanlnu (Noam
Buc Pma. ma Anl: Mt dtsiuawt
\
15) = We - \HNR
U.
a: 9m"
tut Age.
[5: H o cfw_\an
("55: pi Pm ACME],
L': A n (nnm =3
-= Hm 2: pmnm
a Pcnm = yum-Pun
_.
Bmc pmnm 20
etm-prmzo =5 Pun a run \ * Ir
dlui

MA 2621: Probability for Applications
Quiz 3
Feb. 3
Total: 10 pts.
Two fair dice are tossed independently. Let random variable M be the maximum of the two tosses, so
M (1, 5) = 5, M (3, 3) = 3, etc.
1. List all possible values of M and the outcomes associ

MA 2621: Probability for Applications
Quiz 4
Feb. 10
Total: 10 pts.
Let discrete random variable X have the following distribution
X=x
P (X = x)
1
0.2
2
0.4
3
0.3
4
0.1
1. Find the expected value of X. (2 pts.)
Ans:
E(X) =
4
X
xP (X = x)
x=1
= 1 0.2 + 2 0

MA 2621: Probability for Applications
Quiz 5
Feb. 17
Total: 10 pts.
During daytime hours, 60% of all vehicles going across a certain bridge are passenger cars. Let X be the
number of passenger cars among arbitrary 25 vehicles going across the bridge.
1. G

MA 2621: Probability for Applications
Quiz 2
January 27
Total: 10 pts.
Roll two dice. Let A =cfw_the first die is odd, B =cfw_the second die is odd, and C =cfw_the sum is odd.
1. Find the probability P (A). (1 pts.)
Ans: A = cfw_1, 3, 5 cfw_1, 2, 3, 4, 5,

MA 2621: Probability for Applications
Quiz 6
Feb. 24
Total: 10 pts.
Let X be a continuous random variable with density function f (x) = ( 1)x for x 1, where > 1
is a given parameter.
R
1. Show that f (x) = 1. (2.5 pts.)
Ans:
Z
Z
( 1)x dx =
f (x)dx =
1
1

MA 2621: Probability for Applications
Quiz 1
January 20
Consider the experiment of rolling 2 dice, named D1 and D2 , respectively.
1. How many outcomes are there in the sample space generated by the experiment? (2 pts.)
Ans: The sample space = cfw_(x1 , x

Conference Problems: BT Chapter 1
1. A hospital administrator codes incoming patients suffering gunshot wounds according to whether
they have insurance (coding 1 if they do and 0 if they do not) and according to their condition,
which is rated as good (g)