EL6303
HW 7 Solutions
Fall 2015
1. (Book: 6-53) Prove or disprove that, if Ecfw_X 2 = Ecfw_Y 2 = Ecfw_XY , then X = Y in
the MS sense.
Solution: If Ecfw_X 2 = Ecfw_Y 2 = Ecfw_XY , then we have
Ecfw_( X Y )2 = Ecfw_X 2 2Ecfw_XY+ Ecfw_Y 2 = 0
Hence, X =Y in
EL6303
Solutions to HW 10
Fall 2015
1. From Lecture Notes, collect the definitions and simple properties (probability
masses or density, mean, autocorrelation and auto-covariance functions) of
examples of stochastic processes:
(1) Poisson process.
(2) Poi
EL6303
HW 1 Solution
Fall 2015
1. Given: 0 < P( A) <1, 0 < P(B) <1. Prove P( A| B) + P( A| B) =1 iff A, B are
independent.
(Video is required.)
Solution:
" "
P( A| B) + P( A | B) =1 P( AB) + P( AB) =1 P( A) P( AB) + P(B) P( AB) =1
P( B )
P( B)
1 P( B)
P(
EL6303
HW 8 Solutions (Nov 4)
Fall 2015
1. Show that if X (t ) is a WSS process with derivative X '(t ) , then for a given t,
the random variables X (t ) and X '(t ) are orthogonal and uncorrelated.
(Video is required)
Proof: X (t ) is WSS, then Ecfw_X (t
EL6303
Solution to HW 2
Fall 2015
1. The customer flow at a grocery store can be viewed as a Poisson process, i.e.
k
(
)
t
P ( The # of customer arrivals in (0, t ) = k ) = e t
. Suppose the
k!
probability that in two minutes at least one customer arrives
EL6303
Solutions to HW 9
Fall 2015
1. Suppose W (t ) is the Wiener process with Ecfw_W (t ) = 0 and
RW (t1,t2) = min(t1,t2) . Define X (t ) =W (t +1) W (t).
Find Ecfw_X (t ) and RX (t1,t2 ) . Is X (t ) WSS?
(Video is required.)
Solution:
Ecfw_X (t ) = Ecf
EL6303
HW 11 Solution
Fall 2015
1. The stochastic process X (t ) is real WSS with autocorrelation RXX ( ) and power
spectral density S XX ( ) . Show that
(a) RXX ( ) is real and even.
(b) S XX ( ) is even and non-negative.
(Video is required)
Solution: X
EL6303
HW 6 Solutions
Fall 2015
1. X and Y have joint density function
f ( x, y) A(1 xy), | x | 1,| y |1; zero,otherwise.
XY
(1)
(2)
(3)
(4)
Find A so that the above defined is a valid joint density function.
Are X,Y uncorrelated? Show your answer.
Are X,
EL6303
Solution to HW 3
Fall 2015
1. Let P( X = k ) = Ak (1/3) k 1, k =1,2,., .
(a) Find A so that P( X = k ) represents a probability mass function.
(b) Find Ecfw_X .
(c) Find Ecfw_X 2.
(d) Find the conditional probability mass function P( X = k |1 X 10)
EL6303
HW 5 (Due Wed, Oct 7, 5pm)
Fall 2015
1. X and Y are independent with f X (x) = u(x) u(x 1), fY ( y) = u( y 1) u( y 2) .
Z = X +Y . Find f Z (z) by
f X ( ) fY ( z )d
(1) Convolution Method. f Z ( z) = f X ( z) fY ( z) =
(2) Geometry Method.
(Vid
EL6303
Exercise Problems with Solutions
Fall 2015
1. For a Markov chain, we are given the probability transition matrix
1
, where 0 < , <1. Assuming the limit (stationary) distribution
P =
1
exists. Find the limit distribution and limit transition matr
EL6303
HW 1 Solution
Spring 2017
1. For any arbitrary events A, B, C with P(C) 0 , prove or disprove that
P( A B)| C) 1 P( A B C) .
P(C)
Solution:
Theorem: A B P( A) P(B) .
C is a subset of A B C P( A B C) P(C)
P( A B)C) P( A B C) P(C) P( A B)C) P(C) P(
EL6303
Solution to HW 5
1. X and Y are independent with
Z X Y
. Find
f Z ( z)
f X ( x) u( x 1) u( x 2), fY ( y) u( y) u( y 1)
f Z ( z) f X ( z) fY ( z)
f X ( ) fY ( z )d
(2) Geometric Method.
Solution: (1) Convolution approach. Let us calculate
Z X Y
.
b
EL6303
1. Suppose
W (t )
is a stochastic process with
RW (t1,t2 ) min(t1,t2 )
Find
Spring 2017
Solutions to HW 9
Ecfw_Y (t )
and
. Define
RY (t1,t2 )
. Is
Ecfw_W (t ) 0
and
Y (t) W (t ) W (t 1) for t 1; 0 for t 1.
Y (t )
WSS?
(Video is required.)
Solution
EL6303
Solution to HW 3
Spring 2017
1. Let
P( X k ) Bkp k 1, k 1,2,., ; 0 p 1.
(a) Find B so that
represents a probability mass function.
P( X k )
(b) Find
(c) Find
(d) Find
.
2
2
2
Ecfw_X .
Ecfw_X .
Ecfw_X E cfw_X
(e) Let
. Find the conditional probabi
EL6303
HW 4
Solution
Spring 2017
1. Find and draw FY ( y ) and fY ( y ) in terms of FX ( x) and f X ( x) .
P
2, for x 0
0, for 0 x 1
y g ( x)
x 1, for 1 x 2
1, for x 2
fX ( x)
x
(Video is required.)
Solution:
2, for x 0
0, for 0 x 1
y g ( x)
x 1,
EL6303
HW 6 Solutions
Spring 2017
1. X and Y have joint density function
f ( x, y)
XY
(1)
(2)
(3)
(4)
1 xy
A
, | x |1,| y |1; zero,otherwise.
Find A so that the above defined is a valid joint density function.
Are X,Y uncorrelated? Show your answer.
Are
EL6303
Solutions to HW 10
Spring 2017
1. From Lecture Notes, collect the definitions and simple properties (probability
masses or density, mean, autocorrelation and auto-covariance functions) of
examples of stochastic processes:
(1) Poisson process.
(2) P
EL6303
1. Given
Spring 2017
HW 8 Solutions
X (t)
Ecfw_X (t)
(a) Find
is a WSS (wide sense stationary) process with
(constant) and
.
RX (t1, t2 ) RX ( ), where t1 t2
(b) Find
. Let
Y (t ) X '(t )
.
.
Ecfw_Y (t )
Ecfw_X (t )Y (t )
(c) Are X(t) and Y(t) unc
EL6303 Probability and Stochastic Processes
Introduction
Probability and Stochastic processes is an interesting branch of
mathematics that deals with measuring or determining quantitatively the
likelihood that an event occurs. In this course, we will deve
Lecture 2
Chapter 4 The Concept of a Random Variable
4.1 Introduction
Example
In the fair-die experiment, if we assign to the six outcomes fi the
numbers X( fi ) 10i, i.e. X( f1) 10, . X( f 6 ) 60,
then, X( ) is a random variable.
Definition
A random vari
EL6303
HW 1 Solution
1. For any arbitrary events
A, B, C
with
P( A U B)| C ) 1 P( A U B UC )
P(C )
Solution:
Theorem:
A B P( A) P(B)
Spring 2017
P(C ) 0
, prove or disprove that
.
.
C is a subset of A U B UC P( A U B UC ) P(C )
P( A U B)C ) P( A U B UC )