36-217, Fall 2012
Homework 6 Solutions
1. (15 points)
a) For any x in the range of X, we can write
pX (x) = P (X = x) = P (cfw_X = x A) + P (cfw_X = x Ac ) = pX|A (x)P (A) + pX|Ac (x)P (Ac ),
from which we obtain
E[X] =
x xpX (x)
=
=
=
=
pX|A (x)P (A) + p

36-217 - Probability Theory and Random Processes
Second Midterm
November 6, 2012
Your First and Last Name:
Write the answers to the exam problems on a blue book.
Make sure to write your name also on the blue book you turn in.
You may NOT use a calculat

36-217, Spring 2012
Homework 5
Due Thursday October 11
1. Suppose X Geometric(p), with 0 < p < 1.
(a) Show that P (X x) = (1 p)x1 and P (X > x) = (1 p)x for x = 1, 2, . . .
(b) Show that X enjoys the memoryless property: P (X > x1 + x2 |X > x1 ) = P (X >

36-217, Fall 2012
Homework 6
Due October 18
1. Conditional Expectation.
(a) Show that, for any event A,
E[X] = E[X|A]P (A) + E[X|Ac ]P (Ac )
(b) Let X be a random variable. Show that if B is an event of positive probability and A1 , A2 , . . . , An
are di

Probability Theory and Random Processes, Fall 2014
Homework 2
Due September 11
REMEMBER TO WRITE YOUR NAME AND ANDREW ID ON THE FIRST PAGE OF YOUR ASSIGNMENT
(15pts )
1. In a certain school, 60% of the students wear neither a ring nor a necklace, 20% wear

Further Topics on Random Variables
1
Derived Distributions
2
Covariance and Correlation
Carnegie Mellon University
3
Conditional Expectation and Variance Revisited
Fall 2014
4
Transforms
Further Topics on Random Variables
(CMU)
Fall 2014
1 / 32
(CMU)
Deri

36-217: Probability Theory and Random Processes
Fall 2014
Syllabus
Updated August 26, 2014
Instructors:
O ce:
e-mail:
O ce Hours:
Alessandro Rinaldo
Baker Hall 229A
[email protected]
Alex Rojas
Wean Hall 8106
[email protected]
MWF 9:00 to 10:30am or by app

Interpretations of Probability
Sample Space and Probability
1
Interpretations of Probability
2
Set Operations
3
Probability Models
4
Probability Laws
5
Counting
6
Conditional Probability
7
Introduction to Probability
Total Probability Theorem and Bayes Ru

Continuous Random Variables
1
Probability Density Function and Cumulative Distribution Function
2
Expected Value and Variance
Carnegie Mellon University
3
Some Important Continuous Random Variables
Fall 2014
4
Joint PDFs of Multiple Random Variables
Conti

Introduction
Introduction
Given a set of states, S = cfw_s1 , s2 , ., sr . The Markov chain starts in one
of these states and moves successively from one state to another. Each
move is called a step. If the chain is currently in state si , then it moves t

Probability Theory and Random Processes, Fall 2014
Homework 7
Due October 23
REMEMBER TO WRITE YOUR NAME AND ANDREW ID ON THE FIRST PAGE OF YOUR ASSIGNMENT
(15pts )
1. Choose a number X at random from the set of numbers cfw_1, 2, 3, 4, 5. Now choose a num

Probability Theory and Random Processes, Fall 2014
Homework 8
Due October 30
REMEMBER TO WRITE YOUR NAME AND ANDREW ID ON THE FIRST PAGE OF YOUR ASSIGNMENT
(20pts )
1. Let X be a random variable with p.m.f.
pX (x) =
(a)
(b)
(c)
(d)
(e)
(2 pts)
(3 pts)
(5

Probability Theory and Random Processes, Fall 2014
Homework 9
Due November 6
REMEMBER TO WRITE YOUR NAME AND ANDREW ID ON THE FIRST PAGE OF YOUR ASSIGNMENT
(10pts )
1. Consider the two-sided exponential p.d.f.
fX (x) =
pex ,
(1 p)ex ,
if x 0
if x < 0
wher

Probability Theory and Random Processes, Fall 2014
Homework 11
Due November 20
REMEMBER TO WRITE YOUR NAME AND ANDREW ID ON THE FIRST PAGE OF YOUR ASSIGNMENT
(15pts )
1. A simplied model for the spread of a disease goes this way: The total population size

Probability Theory and Random Processes, Fall 2014
Homework 10
Due November 20
REMEMBER TO WRITE YOUR NAME AND ANDREW ID ON THE FIRST PAGE OF YOUR ASSIGNMENT
(15pts )
1. Suppose X and Y have the joint pdf
fX,Y (x, y) =
(15
pts
)
Let S = X + Y . Find the p

Discrete Random Variables
1
Random Variables
2
Important Distributions
3
Bernoulli Process
Carnegie Mellon University
4
Expectation, Mean and Variance
Fall 2014
5
Functions of Random Variables
6
Joint PMFs of Multiple Random Variables
7
Conditioning and I

Probability Theory and Random Processes, Fall 2014
Homework 7
Due October 23
REMEMBER TO WRITE YOUR NAME AND ANDREW ID ON THE FIRST PAGE OF YOUR ASSIGNMENT
(15pts )
1. Choose a number X at random from the set of numbers cfw_1, 2, 3, 4, 5. Now choose a num

Probability Theory and Random Processes, Fall 2014
Homework 1
Due September 4
REMEMBER TO WRITE YOUR NAME AND ANDREW ID ON THE FIRST PAGE OF YOUR ASSIGNMENT
(10pts )
1. Suppose that A B. Show that B c Ac .
(20pts )
2. Suppose that a number x is to be sele

Probability Theory and Random Processes, Fall 2014
Homework 4
Due September 25
REMEMBER TO WRITE YOUR NAME AND ANDREW ID ON THE FIRST PAGE OF YOUR ASSIGNMENT
(10pts )
1. In answering a question on a multiple choice test, a student either knows the answer

= n nE 1
=nn
k=0
1
n
1
M
1
n
k
k
e
k!
1
(1 n ) )k
Probability Theory n e
= n and Random Processes, Fall 2014
Homework 5 k!
k=0
Due (11/n)
October 2
= n ne e
We recognize this as the po
series for an exponential.
REMEMBER TO WRITE YOUR NAME = n n e/nID O

= nE 1 1
1
n
M
1 M
n
Probability Theory and
Random Processes, Fall 2014
1 k k
1
= n Homework 5
n
e
n
k!
Duek=0
October 2
1
(1 n ) )k
We recognize this as the po
= n n e ID ON THE FIRST PAGE OF YOUR ASSIGNMENT
REMEMBER TO WRITE YOUR NAME AND ANDREW
k!
s

Probability Theory and Random Processes, Fall 2014
Homework 6
Due October 16
REMEMBER TO WRITE YOUR NAME AND ANDREW ID ON THE FIRST PAGE OF YOUR ASSIGNMENT
(30pts )
1. Let W be a random variable that takes values from 1 to 8 with equal probability 1/8. De

Probability Theory and Random Processes, Fall 2014
Homework 4
Due September 25
REMEMBER TO WRITE YOUR NAME AND ANDREW ID ON THE FIRST PAGE OF YOUR ASSIGNMENT
(10pts )
1. In answering a question on a multiple choice test, a student either knows the answer

Probability Theory and Random Processes, Fall 2014
Homework 3
Due September 18
REMEMBER TO WRITE YOUR NAME AND ANDREW ID ON THE FIRST PAGE OF YOUR ASSIGNMENT
(20pts )
1. Quick Proofs.
(a) (5 pts) If A and B are disjoint events and Pr(B) > 0, what is the v

Probability Theory and Random Processes, Fall 2014
Homework 1
Due September 4
REMEMBER TO WRITE YOUR NAME AND ANDREW ID ON THE FIRST PAGE OF YOUR ASSIGNMENT
(10pts )
1. Suppose that A B. Show that B c Ac .
Solution: Assume that x B c . We need to show tha

Probability Theory and Random Processes, Fall 2014
Homework 2
Due September 11
REMEMBER TO WRITE YOUR NAME AND ANDREW ID ON THE FIRST PAGE OF YOUR ASSIGNMENT
(15pts )
1. In a certain school, 60% of the students wear neither a ring nor a necklace, 20% wear

Probability Theory and Random Processes, Fall 2014
Homework 3
Due September 18
REMEMBER TO WRITE YOUR NAME AND ANDREW ID ON THE FIRST PAGE OF YOUR ASSIGNMENT
(20pts )
1. Quick Proofs.
(a) (5 pts) If A and B are disjoint events and Pr(B) > 0, what is the v

Probability Theory and Random Processes, Fall 2014
Homework 6
Due October 16
REMEMBER TO WRITE YOUR NAME AND ANDREW ID ON THE FIRST PAGE OF YOUR ASSIGNMENT
(30pts )
1. Let W be a random variable that takes values from 1 to 8 with equal probability 1/8. De