EECS 501. Probability and Random Processes
Fall 2014
Problem Set #4
Issued: 10/2/14, Due: 10/9/14
Problem 1
Trains headed for destination A arrive at the train station at 15-minute intervals starting at
7 a.m., whereas trains headed for destination B arri

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EECS 501 F10 Discussion - Oct. 25, 26
Problem 1 (Transformation of variable)
Determine the pdf of the random variable Y = sin1 (X) when:
(a) X is uniformly distributed on [0, 1]
(b) X is uniformly distributed on [1, 1]
Problem 2
Let X be a uniformly distr

EECS 501 F10 Discussion - Week 2
Problem 1 (Law of total probability)
Suppose that you continually collect coupons and that there are m dierent types. Suppose also
that each time a coupon is obtained it is a type i coupon with probability pi , i = 1, ., m

EECS 501. Probability and Random Processes
Fall 2014
Problem Set #8
Issued: 11/7/14, Due: 11/13/14
Problem 1
Let X and Y be random variables with given joint pdf fXY . We observe Y and wish to form
a linear estimate X of X, i.e.
X = cY + d
so as to minimi

EECS 501. Probability and Random Prosses
Fall 2014
Problem Set #3
Issued: 9/25/14, Due: 10/2/14
Problems 38, 52 in Chapter 2 of the textbook.
Problem 3
Let X be a discrete random variable. Prove that if Variance(X) = 0 then X is almost surely
constant, th

EECS 501. Probability and Random Processes
Fall 2014
Problem Set #9
Issued: 11/20/14, Due: 12/04/14
Problem 1
Problem 33, Chapter 10 of the textbook.
Problem 2
Problem 34, Chapter 10 of the textbook.
Problem 3
Problem 20, Chapter 10 of the textbook.
Probl

EECS 501. Probability and Random Processes
Fall 2014
Problem Set #10
Issued: 12/4/14, Due:
Problem 1
Prove that any random process with independent increments is a Markov process.
Problem 2
An absent-minded professor schedules two student appointments for

EECS 501. Probability and Random Processes
Fall 2014
Problem Set #6
Issued: 10/16/14, Due: 10/28/14
Problem 1
The lifetime of a machine (in days) is a r.v. T with PMF f . Given that the machine is
working after t days, what is the subsequent life of the m

EECS 501. Probability and Random Processes
Fall 2014
Problem Set #5
Issued: 10/9/14, Due: 10/16/14
Problem 1
A total of n bar magnets are placed end-to-end in a line with random independent orientations. Adjacent like poles repel, ends with opposite polar

EECS 501 F10 Discussion - Nov 1, 2
Problem 1 (Gubner 7.32)
Let X and Y be jointly continuous random variables with joint density fxy . Find fZ (z) if
(a) Z = eX Y
(b) Z = |X + Y |
Problem 2
Let X1 , X2 , . . . , Xn be independent random variables dened on

EECS 501 F10 Discussion - 4
Problem 1 An evening event is being attended by n couples. Suddenly k people are asked to leave
the event. Assuming that the k people have been picked at random, compute the expected number
of couples left in the event.
Problem

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EECS 501 F10 Discussion - Nov 15, 16
Problem 1 (Gubner 9.16)
Let X, Y , U and V be jointly Gaussian with X and Y independent N (0, 1). Put
X Y
Z := det
U
V
If [X, Y ] and [U, V ] are uncorrelated random vectors, nd the conditional density fZ|U V (z|u,v)
P

EECS 501 F10 Discussion - Nov 8, 9
Problem 1
Let X and Y be random variables dened on the same probability space. Dene
V ariance(Y |X = x) = E(cfw_Y E(Y |X = x2 |X = x).
Prove that
V ariance(Y ) = Ecfw_V ariance(Y |X) + V ariancecfw_E(Y |X)
Problem 2
Supp

EECS 501 F10 Discussion - Nov 22, 23
Problem 1
Given a WSS process and its autocorrelation function RX ( ) = E(Xt1 Xt1 + ). The Fourier transform of RX ( ) is the power spectrum density function
SX (f ) =
RX ( )ej2f d.
Show that SX (f ) is real and even f

EECS 501 F10 Discussion - Dec 6, 7th
Problem 1
Consider a Poisson process. Suppose we know that there is only one arrival occurred in the time
interval [0, t]. Show that condition on this knowledge, the arrival time of this event is uniform
distributed in

EECS 501 F10 Discussion - Dec 13, 14
Problem 1 A person enters a bank and nd all of the four clerks busy serving customers. There
are no other customers in the bank, so the person will start service as soon as one of the customers
in service leaves. Custo

EECS 501 F10 Discussion - Nov 29, 30
Problem 1
Determine which of the following function cannot be valid power spectrum densities and explain
2
f
(a) f 6 +3f 2 +3
(b)e(f 1)
2
2
4
f
(c) f 4 +1 (f )
f
(d) jf 6 +f 2 +1
Problem 2
If E(X) < 0 and = 0 is such t

EECS 501 F10 Discussion - Week 3
Problem 1
The number of coins in your pocket is a Poisson random variable with parameter . You toss each
coin. Each coin ip is independent and identically distributed with probability p begin head. Show
that the total numb

EECS 501 F10 Discussion - Week 5
Problem 1
X is a continuous positive random variable with pdf f and cdf F . Dene the failure rate
1
r(t) = lim P (X t + h|X > t) where x 0
+ h
h0
f (t)
(a) Show that r(t) =
1 F (t)
(b) Show that failure rate of exponential

EECS 501. Probability and Random Processes
Fall 2014
Problem Set #7
Issued: 10/30/14, Due: 11/6/14
Problem 1
Problem 24, Chapter 7 of the textbook.
Problem 2
Let Y and U be two independent random variables dened on the same probability space
1
(, F, P ).

EECS 501. Probability and Random Prosses
Fall 2014
Problem Set #1
Issued: 9/11/14, Due: 9/18/14
Problems 48, 70, 79, 81, 33, 34, 37, in Chapter 1 of the textbook.
Problem 8
Suppose we repeatedly ip a coin. Successive coin ips are independent. Each coin ip

EECS 501. Probability and Random Prosses
Fall 2014
Problem Set #2
Issued: 9/18/14, Due: 9/25/14
Problems 35, 40, 38, 55 in Chapter 1 of the textbook.
Problem 5
The following experiment is performed. A class of 10 students is surveyed to nd out how
many ow

EECS 501
Discussion 8
1
Joint Continuous Random Variables
Problem 1: For the following two joint pdfs, are the random variables X and Y independent? (Prove your
assertion without directly calculating the marginals.)
2
1)fX,Y (x, y) = e2x3y , (x, y) R+
2)f

EECS 501
Discussion 6
1
Probability Space
2
Combinatorics and Probability
3
The Law of Total Probability and Bayes Rule
4
Discrete Random Variables and Continuous Random Variables
5
Expected Values and Linearity of Expectation
6
Markovs and Chebyshevs Ine

EECS 501
Discussion 11
1
Stationary Random Process
Problem 1 Let cfw_Xn , n = 1, 2, . . . be a strictly stationary random process where each Xn is a discrete
random variable. Define Yn = q(Xn , Xn+1 , . . . , Xn+k1 ), where q(xn , xn+1 , . . . , xn+k1 )