Solution to Homework 1
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.

EL630: Homework 5
1. (Book: 5-2) Find Fy ( y) and f y ( y) if Y = 4 X + 3 and f x ( x) = 2e2 xu( x).
2. (Book: 5-4) The random variable
f y ( y) and Fy ( y) if
X
is uniform in the interval (-2c, 2c) . Find and sketch
Y = X 2.
3. (Book: 5-6) The random var

Solution to HW# 4
1. (Book: 4-9) Find f ( x) if F ( x) = (1 e x )u( x c).
Solution: Applying the sampling property of the impulse function:
h( x) ( x a) = h(a) ( x a), then we have
f ( x) = d F ( x) = d (1 e x )u( x c)
dx
dx
= ( d (1 e x )u( x c) + (1 e x

Solution to HW# 5
1. (Book: 5-2) Find Fy ( y) and f y ( y) if Y = 4 X + 3 and f x ( x) = 2e2 xu( x).
(In this solution, we are going to apply some properties of impulse functions:
d
u( x) = ( x) .
dx
b) ( x)= ( x) (The delta function is an even function.)

EL630: Homework 5
1.
2 x
(Book: 5-2) Find Fy ( y) and f y ( y) if Y = 4 X + 3 and f x ( x) = 2e u( x).
2.
(Book: 5-4) The random variable
f y ( y) and Fy ( y) if
3.
4.
5.
6.
7.
X
is uniform in the interval (-2c, 2c) . Find and sketch
Y = X 2.
(Book: 5-6)

EL630: Homework 4
1. (Book: 4-9) Find f ( x) if F ( x) = (1 e x )u( x c).
F ( x)
c
x
(Hint: For any function h( x), we have h( x) ( x a) = h(a) ( x a), which is called the
sampling property of the impulse function.)
2. (Book: 4-10) If
X
is N (0, 2) find (

Solutions to HW# 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).
Solution: If A B, P( A) = 1/ 4, P(B) = 1/ 3, then
P( A | B) =
P ( AB ) P ( A) 1 / 4
P( AB ) P( A)
=
=
= 3 / 4, P( B | A) =
=
= 1.
P( B)
P( B) 1 / 3
P( A

Lecture 9
6.6 Conditional Distribution
P cfw_ Y y, M
Recall FY ( y | M ) = P cfw_ Y y | M =
Pcfw_ M
Now, there are two RVs: X, Y.
Definition
FY ( y | X = x ) @P cfw_ Y y | X = x is called the conditional distribution
of Y assuming X = x .
fY ( y | X =

Lecture 10
Chapter 7 Sequences of Random Variables
7.1 General Concepts
Definition
The random vector is a vector X = [x1 , x 2 ,L, x n ] whose components
xi s are random variables.
Definition
The joint distribution is
F ( X ) = F ( x1 , x2 ,L, xn ) = P cf

Lecture 12
The System Parameter Estimations
Example 1
X i +1 = aX i +Vi +1, with X 0 = 0,
(1)
with V = 0, V = 1 ,
where Vi , i = 1,2,., N , is an i.i.d normal sequence,
and a is
i
i
an unknown parameter to be estimated based on the observations
2
X i , i

Lecture 3
Independence of Three Events
Definition
A1 , A2 , A3
and
are called independent if they are independent in pairs:
P ( Ai A j ) = P ( Ai ) P ( A j )
i j
P ( A1 A2 A3 ) = P ( A1 ) P ( A2 ) P ( A3 )
Note
Three events might be independent in pairs b

Solution to HW#6
1. (Book: 5-26) For a Poisson random variable X with parameter show that
1
2
(a) P(0 < X < 2 ) >
; (b) E[ X ( X 1)] = , E[ X ( X 1)( X 2)] = 3.
Solution: (a) X is a Poisson random variable with parameter means that :
k
P ( X =k ) = e
.

Lecture 5
4.5 Asymptotic Approximations for Binomial Random Variable
A
A
P ( A) = p, P ( A) = q, p + q = 1
n
Pn (k ) = Pcfw_ A occurs k times in n trials = p k q nk
k
k
n
Pcfw_ A occurs k1 to k2 times in n trials = p k q nk
k
k =k
The Normal Approximat

Lecture 4
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 ( f i ) = 10i,
then, X( ) is a random variable.
i.e. X( f1) = 10, . X( f 6 ) = 60,
Definition
A ran

EL630: Homework 7
1. (Book: 6-1) X and Y are independent, identically distributed (i.i.d.) random variables
with common p.d.f. (probability density function)
y
f ( x) =e xu( x), f ( x) =e u( y)
X
Y
Find the p.d.f. of the following random variables
(a) X +

EL630: Homework 8
1. (Book: 6-50) Show that if X and Y are independent exponential random variables with
y
f ( x) =e xu( x), f ( x) =e u( y)
X
Y
and Z = ( X Y )u( X Y ) , then Ecfw_Z =1/ 2 .
2. (Book: 6-51) Show that for any X , Y real or complex (a) | Ec

Solution to HW# 8
1.
(Book: 6-50) Show that if X and Y are independent exponential random variables with
y
f ( x) =e xu( x), f ( x) =e u( y) and Z = ( X Y )u( X Y ) , then Ecfw_Z = 1/ 2 .
X
Y
y
Solution: Ecfw_Z = ( x y)u( x y) f ( x, y)dxdy = ( x y)u( x y

EL630: Homework 9
1.
(Book: 6-20) X and Y are independent exponential random variables with
f ( x) = e xu( x), f ( x) = e yu( y)
X
Y
5.
Find the densities of the following random variables:
(a) 2X+Y (b) X-Y (c) X/Y (d) max(X,Y) (e) min(X,Y)
(Book: 6-52) S

Solution to HW#9
1.
(Book: 6-20) X and Y are independent exponential random variables with
f ( x) = e xu( x), f ( x) = e yu( y)
X
Y
Find the densities of the following random variables:
(a) 2X+Y (b) X-Y (c) X/Y (d) max(X,Y) (e) min(X,Y)
Solution: (a) Z=2X

EL630: Homework 10
1.
(Book: 7-6) Show that if the random variables X, Y and Z are such that
r = r =1, then r =1.
XY
2.
3.
4.
5.
6.
YZ
XZ
(Book: 7-25) Show that if an a and Ecfw_| X n an |2 0, then X n a in MS sense as
n.
f ( x, y ) = 1/ , x2 + y 2 1; zer

Solution to HW#10
1.
(Book: 7-6) Show that if the random variables X, Y and Z are such that r = r = 1,
XY
then r = 1.
YZ
XZ
Solution: From Problem 6-52 (in HW#9): Y =aX +b, Z =cY + d , hence,
Z = AX + B
= A + B
= A
Z
X
Z
X
Ecfw_( Z )( X ) = Ecfw_A( X )(

EL630: Homework 11
1.
Let X1, X 2,⋅⋅⋅, X n be independent, identically distributed normal random variables with
common p.d.f.
1
f X ( xi ;µ ) =
2πσ 2
i
e
−
( xi − µ ) 2
2σ 2
where µ is the only unknown parameter.
ˆ
(a) Find µ the ML estimate for µ .
ˆ
(b)

EL630: Homework 11
1.
Let X1, X 2, X n be independent, identically distributed normal random variables with
common p.d.f.
1
f X ( xi ; ) =
2 2
i
e
( xi ) 2
2 2
where is the only unknown parameter.
(a) Find the ML estimate for .
(b) Show that is an unbiase

EL630: Homework 12
1. For the system in Example 1 of Lecture 12:
X i +1 = aX i +Vi +1, with X 0 = 0,
show that if
(1)
N 1
X iVi +1
i =0
N N 1
lim
= 0, w.p.1.
2
Xi
i =0
then
N 1
a=
X i X i +1
i =0
N 1
a, as N , w.p.1.
2
Xi
i =0
2. For the scalar case

Lecture-11
Principles of Parameter Estimation
The purpose of this lecture is to illustrate the usefulness of
the various concepts introduced and studied in earlier
lectures to practical problems of interest. In this context,
consider the problem of estima

Solutions to HW#7
1. (Book: 6-1) X and Y are independent, identically distributed (i.i.d.) random variables
with common p.d.f. (probability density function)
f ( x ) = e x u ( x), f ( y ) = e y u ( y )
X
Y
Find the p.d.f. of the following random variables

EL630: HW 12 Solution
1. For the system in Example 1 of Lecture 12:
X i +1 = aX i +Vi +1, with X 0 = 0,
show that if
(1)
N 1
X iVi +1
i =0
N N 1
lim
= 0, w.p.1.
2
Xi
i =0
then
N 1
a=
X i X i +1
i =0
N 1
a, as N , w.p.1.
2
Xi
i =0
Solution:
N 1
a=
X

EL6303
Solutions to HW 2
Sep 15, 2013
1. Make up 3 new probability distributions (each distribution should have a
different mathematical basis, such as use an exponential term in one of your
distributions, a linear function in another one of your distribu