ECSE-2500 Engineering Probability HW#1 Solutions
Due 1/28/16
Graded by Qiushi Gong, gongq@rpi.edu
1. (8 points, 1 point each) Let
universal set is
A 1 ,3 , B 1 , C 4 ,6 , D 9 ,
,2 ,4
,5 ,7
,10
where the
U 1 ,3 ,6 ,8 ,10 .
,2 ,4,5 ,7 ,9
1.a. Fill in the b

ECSE-2500 Engineering Probability HW#3 Solutions
Due 2/4/16
Graded by Yin Li liy34@rpi.edu]
[
.
1. (10 points, 2 points each) A number is selected at random in the interval [1,1]. Let
A 0, B 0.3, and C 0.2. Compute P A, P B, P C , P AB,
and P ABC .
Soluti

Sequential Experiments
(Conclusion)
Section 2.6, except 2.6.5.
9/21/2015
ECSE-2500 Lecture 6a: Sequential Experiments End
1
Definition
A sequential experiment is a series of experiments that
together make up one large experiment.
Let E1, E2, , En be n e

ECSE-2500 Engineering Probability HW#9 Solutions
9/29/14
1. (8 points) A single fair 4-sided die is tossed once. Let the random variable X1 = the number of
dots on the face of the die.
1.a. (4 points) Write out the PMF pX1 k and the CDF FX1 x of this RV.

ECSE-2500 Engineering Probability HW#8 Solutions
Due 9/25/14
1. (26 points) For Problem 2 in HW #6, we defined a random variable (on the sample space defined
in problem 1 of HW#6) by X2 i, j i j. We then calculated the probability mass function
(PMF) and

ECSE-2500 Engineering Probability HW#9
9/29/14
1. (8 points) A single fair 4-sided die is tossed once. Let the random variable X1 = the number of
dots on the face of the die.
1.a. (4 points) Write out the PMF pX1 k and the CDF FX1 x of this RV. Graph the

ECSE-2500 Engineering Probability HW#7 Solutions
Due 9/22/14
1. (4 points) In Problem 1 of HW #6, we had the following set up: Two distinct fair dice are tossed
independently. We record the number on die 1 followed by the number on die 2 as an ordered
pai

ECSE-2500 Engineering Probability HW#8
Due 9/25/14
1. (26 points) For Problem 2 in HW #6, we defined a random variable (on the sample space defined
in problem 1 of HW#6) by X2 i, j i j. We then calculated the probability mass function
(PMF) and computed E

ECSE-2500 Engineering Probability HW#6 Solutions
Due 9/18/14
1. (12 points) Two distinct fair dice are tossed independently. We record the number on die 1
followed by the number on die 2 as an ordered pair, so that our sample space is
S s1 , s2 si1,2,3,4,

ECSE-2500 Engineering Probability HW#10
10/2/14
1. (8 points) A single fair 4-sided die is tossed once. Define the random variable X2 = one over the
number of dots on the face of the die. In HW#9, you found the PMF pX2 k and the CDF
FX x of this RV.
2
1.a

ECSE-2500 Engineering Probability HW#11
Due 10/6/14
1. (4 points) A single fair 4-sided die is tossed once. Define the random variable X2 = one over the
number of dots on the face of the die. In HW#9, you found the PMF pX2 k and the CDF
FX x of this RV. I

ECSE-2500 Engineering Probability HW#13 Solutions
Due 10/20/14
1. (12 points) Let X be a Gaussian random variable with mean 5.5 and variance one representing
the height of an adult chosen at random (measured in feet).
1.a. (3 points) Use the Markov Inequa

ECSE-2500 Engineering Probability HW#14 Solutions
Due 10/23/14
1. (10 points) Let X and Y be discrete RVs with joint PMF pX,Y x, y given in the following table:
pX,Y x, y
1
x
0
1
pY y
1
0.06
0.14
0.10
0.30
y
0
0.04
0.05
0.03
0.12
pX x
0.45
0.40
0.15
1

ECSE-2500 Engineering Probability HW#14 Solutions
Due 10/23/14
1. (10 points) Let X and Y be discrete RVs with joint PMF pX,Y x, y given in the following table:
pX,Y x, y
x
1
0
1
y
1
0
1
0.06 0.04 0.35
0.14 0.05 0.21
0.10 0.03 0.02
1.a. (4 points) Comput

ECSE-2500 Engineering Probability HW#12 Solutions
Due 10/16/14
1. (6 points, 3 points each part) Let X have a Gaussian distribution with mean 1 and variance 16.
1.a. Let Y 2X 4. What is the PDF of Y?
Solution
This Y g X 2X 4 is affine, so we can apply the

ECSE-2500 Engineering Probability HW#13
Due 10/20/14
1. (12 points) Let X be a Gaussian random variable with mean 5.5 and variance one representing
the height of an adult chosen at random (measured in feet).
1.a. (3 points) Use the Markov Inequality to co

ECSE-2500 Engineering Probability HW#12
Due 10/16/14
1. (6 points, 3 points each part) Let X have a Gaussian distribution with mean 1 and variance 16.
1.a. Let Y 2X 4. What is the PDF of Y?
1.b. Let Y eX. What is the PDF of Y?
1
. Use the general
X
formul

ECSE-2500 Engineering Probability HW#11 Solutions
Due 10/6/14
1. (4 points) A single fair 4-sided die is tossed once. Define the random variable X2 = one over the
number of dots on the face of the die. In HW#9, you found the PMF pX2 k and the CDF
FX x of

ECSE-2500 Engineering Probability HW#10 Solutions
10/2/14
1. (8 points) A single fair 4-sided die is tossed once. Define the random variable X2 = one over the
number of dots on the face of the die. In HW#9, you found the PMF pX2 k and the CDF
FX x of this

ECSE-2500 Engineering Probability HW#7
Due 9/22/14
1. (4 points) In Problem 1 of HW #6, we had the following set up: Two distinct fair dice are tossed
independently. We record the number on die 1 followed by the number on die 2 as an ordered
pair, so that

ECSE-2500 Engineering Probability HW#6
Due 9/18/14
1. (12 points) Two distinct fair dice are tossed independently. We record the number on die 1
followed by the number on die 2 as an ordered pair, so that our sample space is
S s1 , s2 si1,2,3,4,5,6, i 1,2

ECSE-2500 Engineering Probability HW#5 Solutions
Due 9/15/14
1. (8 points) Our chip manufacturing facility ships chips in boxes of 100 chips. Assume that
defective chips are independent from one chip to the next and that on average one chip in 100 is
defe

Computing Probabilities
Using Counting Methods
(aka Combinatorics)
Section 2.3 except 2.3.5.
9/10/2015
ECSE-2500 Lecture 3b: Counting Methods
1
Introduction
Here we briefly cover some basic Combinatorics.
Some of you have seen this before in Introductio

Reminder: Exam 1
Exam 1 will be Thursday, October 1st in class.
It will cover all the material through Thursday,
September 24th and HW # 7.
Monday, September 28th will be a review class.
All exams are closed book and notes.
You may bring one sheet of

Probability Basics
Part 2
Read Chapter 2.
(You should already know basic Set Theory.
If not, study Subsection 2.1.3.)
9/3/2015
ECSE-2500 Lecture 2: Basics: Prob
1
Recall Random Experiments
A random experiment is an experiment where the
outcome varies unp

Discrete
Random Variables
Section 3.1, except 3.1.1.
Section 3.2.
9/21/2015
ECSE-2500 Lecture 6b: Discrete Random Variables
1
Recall Bernoulli Trials 1
The sample space for the i th trial is Si h, t
For a fixed n, the sample space for the combined
exper

Probability Basics:
Continuous Spaces
Read Chapter 2.
9/10/2015
ECSE-2500 Lecture 3a: Prob on Continuous Spaces
1
Recall Event Classes and Fields
In some cases (e.g. continuous sample spaces), we do
not want to consider all events.
We create an event cl

Sequential
Experiments
Section 2.6, except 2.6.5.
9/17/2015
ECSE-2500 Lecture 5b: Sequential Experiments
1
Definition
A sequential experiment is a series of experiments that
together make up one large experiment.
Let E1, E2, , En be n experiments with c

Conditional Probability
Section 2.4.
9/14/2015
ECSE-2500 Lecture 4a: Conidtional Prob
1
Partial Information
In engineering problems where we use Probability, we
usually have some partial information that is related
to what we want to know or some decisio

ECSE-2500 Engineering Probability HW#1
Due 8/28/14
1. (12 points) There are three balls in an urn numbered 1,2,3. Hint: You should get different
answers for each part below.
1.a. (4 points) You randomly select one ball from the urn. What is the smallest s

Probability Basics
Part 1
Read Chapter 2.
(You should already know basic Set Theory. If
not, study Subsection 2.1.3.)
8/31/2015
ECSE-2500 Lecture 1c: Basics
1
Basic Set Theory 1
You should know basic ideas from set theory like
Membership: a A or a A
Set

The Importance of
Probability in Engineering
Read Chapter 1 and start Chapter 2
8/31/2015
ECSE-2500 Lecture 1b: Intro
1
What is Probability?
Probability is the study and analysis of randomness
and the search for order in random events and
occurrences.
R

Name: _
RIN#: _
Exam 1
Engineering Probability
ECSE-2500
February 16, 2017
Points
Rules:
1
20
2
25
3
20
4
20
5
15
Total
100
Score
Show all work.
You have 75 minutes to take this exam.
You are allowed one 8.5 x 11 inch piece of paper written on both sides.

ECSE-2500 Engineering Probability HW#9 Solutions
10/6/16
Graded by Usama Munir Sheik [sheiku@rpi.edu]. Show your work.
1. (8 points) A single fair 4-sided die is tossed once. Define the random variable X2 = one over the
number of dots on the face of the d

ECSE-2500 Engineering Probability HW#8 Solutions
10/3/16
Graded by Sagnik Nath [naths3@rpi.edu]. Clearly justify your answer in each problem.
1. (8 points) A single fair 4-sided die is tossed once. Let the random variable X1 = the number of
dots on the fa