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 (Elza Erkip)
HW 2
Fall 2015
1. F ( x) = A(1 ebx )u( x a) , where u( x) is a step function. a and b are positive
constants.
(1) Find constant A so that F(x) is a probability distribution function.
(2) Draw F(x).
(3) Find and draw f ( x) .
(Video is

EL6303 HW 1 (Due 5pm, Wed, Sep 9, 2015)
1. Given:
independent.
Prove
Submit to EL6303F15@gmail.com
Fall 2015
iff A, B are
(Video is required.)
2. A digital signal 1 or 0 is transmitted through a noisy channel, the
received data may be different from the s

EL6303
HW 8 Solution
1. The random process
X (t ) e
at
is a family of exponentials dependending on the
random variable a.
(a) Draw three sample functions of this random process.
(b) Express the mean
, the autocorrelation
, and the first-order
(t )
densit

EL630: Homework 2
1. (Book:2-8) If A B, P( A) 1/ 4, and P( B) 1/ 3, find P( A | B) and P( B | A). = =
2. (Book:2-9) Show that P( AB | C ) = P( A | BC ) P( B | C ) an P ( ABC ) = P ( A | BC ) P ( B | C ) P (C ). d
3. (Book:2-12) A call occurs at time t , w

EL630: Homework 3
1. (Book:3-1) Let p represent the probability of an event A. What is the probability that (a) A occurs at least twice in n independent trials; (b) A occurs at least thrice in n independent trials? 2. (Book:3-2) A pair of dice is rolled 5

EL 6303 Homework 1
Prof. Rappaport
Assigned 9/4/13
Due 9/11/13 beginning
pp. 1-70
1. Define carefully the following words: a) randomness; b) causality; c) axiomatic; d) postulate; e) theory;
f) deductive reasoning; g) induction; h) hypothesis; and i) expe

EL6303
HW 2 (Due Sep 15, Wed, 5pm)
Fall 2015
1. The customer flow at a grocery store can be viewed as a Poisson process, i.e.
Suppose the
k
t t
P The # of customer arrivals in (0, t ) k e
k!
.
probability that in two minutes at least one customer arrive

EL6303
1. Given:
HW 1 Solution
0 P( A) 1, 0 P(B) 1.
independent.
Solution:
" "
Prove
Fall 2015
P( A| B) P( A| B) 1
iff A, B are
(Video is required.)
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( B)
P( A) P(B) P(

Mean Autocorrelation Autocovariance Crosscorrelation Crosscovariance
Definiti n: We say that a real stochastic process X(t) is stationam in strict sense
($5.5. strict sense stationary) if its statistics is not affected by any shift on the
Definition:
The

EL6303
Solutions to HW 9
1. Suppose
W (t )
is the Wiener process with
RW (t1,t2 ) min(t1,t2 )
Find
Ecfw_X (t)
and
. Define
RX (t1,t2 )
Ecfw_W (t ) 0
X (t ) W (t 1) W (t )
. Is
X (t )
Fall 2015
and
.
WSS?
(Video is required.)
Solution:
Ecfw_X (t) Ecfw_W (t

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
Fall 2015
Solutions to HW 10
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 8 Solutions (Nov 4)
1. Show that if
is a WSS process with derivative
X (t )
the random variables
X (t )
and
X '(t )
Fall 2015
, then for a given t,
X '(t)
are orthogonal and uncorrelated.
(Video is required)
Proof:
is WSS, then
Ecfw_X (t )
.
d

EL6303
HW 10 Solution
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 positive.
Solution: X (t ) is real WSS.
RXX ( ) RXX (t ,

EL6303
HW 3 Solution
1. A fair coin is tossed 10,000 times. Find the probability that the number of
heads is between 4,800 and 5,200.
Solution: We know that
if
,
k2 n
npq 1
k nk
k2 np
k1 np
) G(
)
p q G(
k k1
k
are of the order of
npq
npq
.
k1 np and k

EL6303
HW 4 Solution
3pm
1. Let
P( X k ) Ak (1/ 2)k 1, k 1,2,., .
(a) Find A so that
represents a probability mass function.
P( X k )
(b) Find
Ecfw_X .
Solution: Lets discuss a more general problem
P( X k ) Akpk 1, k 1,2,., , with p 1.
,
.
1
k
p 1 p p2 .

EL6303
HW 6 Solutions
1. X and Y are i.i.d. random variables with common p.d.f.
.
f ( x) e xu( x), f ( y) e yu( y)
X
Y
f X ( x)
f Y ( y)
y
x
Find the p.d.f. of the following random variables
(a) X Y , (b) X Y , (c) XY , (d) X / Y .
Solution: X,Y are i.i.d

EL6303
HW 2 Solution
1. (Book 4-1) Suppose that
is,
F ( xu ) u
Solution:
. Show that if
xu
is the u percentile of the random variable X that
f ( x) f ( x)
f ( x)
1 F ( x)
x
1
xu
, then
x1u xu
F ( x)
.
xu
1 u
u
x1u
x
x1u
xu
u
1 u
1
u
is an even function
.

EL6303
HW 1
Solution
1. (Book :2 1) Show that ( a) A B A B A;
(b) ( A B ) AB AB B A .
Solution : (a) According to De Morgan's Law A B AB, we have
A B AB AB ,and A B AB
then , A B A B AB AB A( B B ) AS A.
(b) ( A B ) AB ( A B )( A B ) AA AB B A BB
cfw_ AB

EL6303
HW 7 Solutions
1. (Book: 6-53) Prove or disprove that, if
the MS sense.
Solution: If
Ecfw_X 2 Ecfw_Y 2 Ecfw_XY ,
Fall 2015
Ecfw_X 2 Ecfw_Y 2 Ecfw_XY ,
then X Y
in
then we have
Ecfw_( X Y )2 Ecfw_X 2 2Ecfw_XY Ecfw_Y 2 0
Hence,
X Y
in the MS sense.

Lecture 4
Example Y g ( X ) .
y g ( x)
1
1
1
x
1
Find FY ( y ) and fY ( y ) in terms of FX ( x) and f X ( x) .
1, y 1
Solution: We see that | Y | 1. Thus FY ( y )
0, y 1.
Now for | y | 1, we have y x , thus
FY ( y ) Pcfw_Y y Pcfw_ X y FX ( y ) .
1
FX (

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 5
Chapter 6: Two Random Variables
Suppose A and B are two events. We know that in order to study A
and B, just knowing P ( A) and P ( B) is not enough. We have to know
how they are related to each other. That is we have to know P ( AB).
Similarly,

19. Series Representation of Stochastic Processes
Given information about a stochastic process X(t) in 0 t T , can this continuous information be represented in terms of a countable set of random variables whose relative importance decrease under some arr

18. Power Spectrum
For a deterministic signal x(t), the spectrum is well defined: If X ( ) represents its Fourier transform, i.e., if X ( ) = x(t )e j t dt ,
+
(18-1)
then | X ( ) |2 represents its energy spectrum. This follows from Parsevals theorem sinc

16. Mean Square Estimation
Given some information that is related to an unknown quantity of interest, the problem is to obtain a good estimate for the unknown in terms of the observed data. Suppose X 1 , X 2 , , X n represent a sequence of random variable

15. Poisson Processes
In Lecture 4, we introduced Poisson arrivals as the limiting behavior
of Binomial random variables. (Refer to Poisson approximation of
Binomial random variables.)
From the discussion there (see (4-6)-(4-8) Lecture 4)
" k arrivals occ

Probability 2 - Notes 9
The results for two random variables are now extended to n random variables.
Joint p.d.f. This is defined to be a function fX1 ,X2 ,.,Xn (x1 , ., xn ) such that for any measurable
set A contained in n ,
Z
P(X1 , ., Xn ) A) =
Z
.
(x

Probability 2 - Notes 10
Some Useful Inequalities.
Lemma. If X is a random variable and g(x) 0 for all x in the support of fX , then P(g(X)
1) E[g(X)].
Proof. (continuous case)
Z
P(g(X) 1) =
Z
x:g(x)1
fX (x)dx
x:g(x)1
g(x) fX (x)dx
Z
g(x) fX (x)dx = E

Probability 2 - Notes 13
Continuous n-dimensional random variables
The results for two random variables are now extended to n random variables.
Definition Random variables X1 , .Xn are said to be jointly continuous if they have a joint
p.d.f. which is def

Probability 2 - Notes 5
Continuous Random Variables
Definition. A random variable X is said to be a continuous random variable if there is a function
fX (x) (the probability density function or p.d.f.) mapping the real line into [0, ) such that
R
for any

Probability 2 - Notes 12
The Law of Large Numbers (LLN)
The LLN is one of the most important results of the classical probability theory. We shall
discuss here the so called weak form of this Law.
Theorem 1. Let X1 , X2 , . be a sequence of i.i.d. random