EE 178/278A
Probabilistic Systems Analysis
Handout #8
Nov 30, 2011
Homework #8
Due Wednesday Dec 7
1. Gambling
Let Xn be the amount you win on the nth round of a game of chance. Assume that
X1 , X2 , . . . , Xn are i.i.d. with nite mean E(X) and variance
EE178/EE278A
Probabilistic Systems Analysis
Handout #23
Thursday, December 7, 2012
Homework #8 Solutions
1. (10 points)
a. Recall the transform of X is MX (s) = es
Then,
2 2 /2
.
E esY = E es
k=0
Xk rk
esXk r
=E
k
k=0
MX srk = e
=
r 2k
k=0
2
s2 2
E esXk
EE 178/278A
Probabilistic Systems Analysis
Handout #22
June 1, 2011
Homework #8 Solutions
1
1. (10 points) We know n n Xi E(X) by the weak law of large numbers, which means
i=1
P(|Sn | > ) 0 as n . The limiting value of P(Sn < ) depends on . When
2
< 0,
EE178/EE278A
Probabilistic Systems Analysis
Handout #21
Thursday, November 29, 2012
Homework #7 Solutions
1. (10 points) By inspection, p = 0.5. The discrete r.v. Y is then equal to 1, with
probability 0.5 and to 2, with probability 0.5. Its mean E(Y ) =
EE178/278A: Probabilistic Systems Analysis, Spring 2014
Homework 1
Due April 10, 10:45am
1. Sample space and events
Consider the sample space of all outcomes from ipping a coin 4 times.
(a) List all the outcomes in . How many are there?
(b) Let A be the e
EE 178 Probabilistic Systems Analysis Homework #2 Due Thursday, January 24, 2008
Handout #2 January 17, 2008
1. Catching the train. The probability that Riddley Walker goes for a run in the morning before work is 2/5. If he runs then the probabilit
Lecture Notes 6
Limit Theorems
Motivation
Markov and Chebyshev Inequalities
Weak Law of Large Numbers
The Central Limit Theorem
Condence Intervals
Corresponding pages from B&T: 380385, 388392.
EE 178/278A: Limit Theorems
Page 6 1
Motivation
One of the ke
EE 178/278A
Probabilistic Systems Analysis
Handout #3
Oct 12, 2011
Homework #3
Due Wednesday, Oct 19, 2011
1. Liars. Of the 100 people in a village, 50 always tell the truth, 30 always lie, and 20 always
refuse to answer. A sample of 30 people are drawn w
Lecture Notes 2
Random Variables
Denition
Discrete Random Variables: Probability mass function (pmf)
Continuous Random Variables: Probability density function (pdf)
Mean and Variance
Cumulative Distribution Function (cdf)
Functions of Random Variables
Cor
Lecture Notes 1
Basic Probability
Set Theory
Elements of Probability
Conditional probability
Sequential Calculation of Probability
Total Probability and Bayes Rule
Independence
Counting
EE 178/278A: Basic Probability
Page 1 1
Set Theory Basics
A set is a
EE 178/278A
Probabilistic Systems Analysis
Handout #2
Oct 5th, 2011
Homework #2
Due 5pm, Wednesday, Oct 12th 2011
1. Testing for a disease. The probability that a man has a particular disease is 1/20. John
is tested for the disease but the test is not tot
EE 178/278A
Probabilistic Systems Analysis
Handout #1
September 28, 2011
Homework #1
Due October 5, 2011
Announcement: The homework is due next Wednesday before 5pm. You can
put it in the Homework in box in the EE 178 drawer of the class le cabinet on
the
EE 178/278A
Probabilistic Systems Analysis
Handout #6
Wedneday, Oct 19, 2011
Homework #4
Due Monday Oct 31 5pm Hard deadline
1. Probabilities from cdf
The cdf of random variable X is given by
FX (x) =
1
3
2
+ 3 (x + 1)2
1 x 0
x < 1
0
a. Find the probabili
EE 178/278A
Probabilistic Systems Analysis
Handout #9
No 9, 2011
Homework #6
Due Wednesday, Nov 16
1. Uncorrelation vs. Independence.
Let X and Y be random variables with joint pdf
fX,Y (x, y) =
c if 0 |x| |y|, 0 |y| 1
0 otherwise,
where c is a constant.
EE 178/278A
Probabilistic Systems Analysis
Handout #8
Nov 16, 2011
Homework #7
Due Wednesday, Nov 30
1. MSE estimation
Let Exp(a) and let the number of packets arriving per unit time at a node in a
communication network, X, given , be Poisson with rate .
EE 178/278A
Probabilistic Systems Analysis
Handout #2
October 5, 2011
Homework Solutions #1
1. (15 points)
The Venn diagrams given in Fig. 1,2,3 and 4, for parts (a),(b),(d) and (e), respectively,
verify the relations. For part (c), one can express P(F G)
Lecture Notes 3
Multiple Random Variables
Joint, Marginal, and Conditional pmfs
Bayes Rule and Independence for pmfs
Joint, Marginal, and Conditional pdfs
Bayes Rule and Independence for pdfs
Functions of Two RVs
One Discrete and One Continuous RVs
EE 178/278A
Probabilistic Systems Analysis
Handout #8
Nov 2, 2011
Homework #5
Due Wednesday, Nov 9
1. Two continuous r.v.s
Let X and Y be two continuous random variables having the joint pdf
fX,Y (x, y) =
c y 0, |x| + y 1
0 otherwise.
a. Find c and the ma
EE 178/278A
Probabilistic Systems Analysis
December, 2011
Handout #10
Sample Final Problems
Some additional problems taken from old nal examinations and other places.
1. Inequalities
Label each of the following statements with =, , or NONE. Label a statem
Lecture Notes 5
Transforms
Denition
Examples
Moment Generating Properties
Inversion
Transforms of Sums of Independent RVs
Corresponding pages from B&T: 210219, 232236.
EE 178/278A: Transforms
Page 5 1
Denition
The transform associated with the probabilit
Review Session 2
EE178/278A
Oct 12th
Outline
Conditional probability
Independence
Counting
Homework Hints
Conditional probability
Sequential calculation of probabilities
See lecture notes 1 slides 26 - 28
In general, how many leaf nodes does the tree have
Review Session 8
EE178/278A
Nov 8th
Moment generating function
[B & T 4.1]: Let X be a random variable that takes values 1, 2, and 3
with the following probabilities:
1
2
1
P(X = 2) =
4
1
P(X = 3) =
4
Find the moment generating function and use it to comp
Lecture Notes 7
Random Processes
Denition
IID Processes
Bernoulli Process
Binomial Counting Process
Interarrival Time Process
Markov Processes
Markov Chains
Classication of States
Steady State Probabilities
Corresponding pages from B&T: 271281, 313340
Review Session 3
EE178/278A
Oct 19th
Outline
Counting
Discrete Random Variables
Continuous Random Variables
Geometric Random Variables
Homework Hints
Counting
Extra problems: Alice and Bob each has a deck of playing cards.
Each turns over a randomly selec
Review Session 1
EE178/278A
Oct 5th
Outline
Questions
Examples of sample space, events, probabilities
Basic Probability identities
Conditional probability
Examples of sample space, event, probabilities
Recall from lecture:
Sample space: set of all possibl
Review Session 4
EE178/278A
Oct 26th
Outline
Linearity of expectation
CDFs
Derived CDF
Homework Hints
Expecation
Linearity of expectation: Let (X1 , X2 , X3 , . . . , Xn ) be random
variables that has a joint pmf or pdf; i.e.
(X1 , X2 , X3 , . . . , Xn )
Review Session 7
EE178/278A
Nov 8th
Extra problems
[B & T 3.13]: Let X be a random variable with the following pdf
x
4
fX (x) =
if 1 x 3
otherwise
0
Let A be the event cfw_X 2. Find E(X), P(A), fX|A (x), E(X|A).
1. E(X) = 27/12 1/12 = 13/6
2. P(A) =
3
2
x
Lecture Notes 4
Expectation
Denition and Properties
Covariance and Correlation
Linear MSE Estimation
Sum of RVs
Conditional Expectation
Iterated Expectation
Nonlinear MSE Estimation
Sum of Random Number of RVs
Corresponding pages from B&T: 81-92, 94-98, 1
Review Session 6
EE178/278A
Nov 8th
Outline
Extra problems
Method of indicators
Linear estimation
Homework hints
Extra problems
HW5 Extra problem 1
1. f (y|s): Suppose S = s is xed. Then, Y is simply Z shifted by a
constant. Hence,
FY (y|S = s) = PZ (Z +