Moments of Random Variables
Chapter 3
Section 3.2: Expected Values
3.1
We are given the triangular PDF.
fX ( x )
1
2
0
x
4
2
We have that
x
4
fX ( x ) =
x
1 4
0
0x<2
2x<4
otherwise
Thus,
E[X] =
xf X ( x ) dx =
2
0
x
x  dx +
4
3 2
4
x
x
x 1  dx =
CHAPTER 4
Special Probability Distributions
Section 4.3: Binomial Distribution
4.1
The probability of a six on a toss of a die is p = 1 6 . Let N(4) be a random variable that
denotes the number of sixes that appear in tossing the four dice. Since the outc
Functions of Random Variables
Chapter 6
Section 6.2: Functions of One Random Variable
6.1
Given that X is a random variable and Y = aX b , where a and b are constants, then
y+b
y+b
F Y ( y ) = P [ Y y ] = P [ aX b y ] = P X  = F X 
a
a
fY ( y ) =
1
y+
Multiple Random Variables
Chapter 5
Section 5.3: Bivariate Discrete Random Variables
5.1
kxy
p XY ( x, y ) =
0
a.
otherwise
To determine the value of k, we have that
x
b.
x = 1, 2, 3 ; y = 1, 2, 3
3
p XY ( x, y ) = 1 = k
y
3
x = 1y = 1
3
xy = k
x=1
1
x
Introduction to Random Processes
Chapter 8
Section 8.3: Mean, Autocorrelation Function, and Autocovariance Function
8.1
Since the function
X(t) = A
0tT
is an aperiodic function, its autocorrelation function is given by
R XX ( t, t + ) =
X ( t )X ( t + ) d
Introduction to Statistics
Chapter 11
Section 11.2: Sampling Theory
11.1
A sample size of 5 results in the sample values 9, 7, 1, 4, and 6.
a.
9+7+1+4+6
The sample mean is X =  = 27 = 5.4
b.
The sample variance is given by
5
1
2
S = 5
5
(X
k
5
1
2
2
2
Transform Methods
Chapter 7
Section 7.2: Characteristic Functions
7.1
We are given a random variable X with the following PDF:
1
fX ( x ) = b a
0
a<x<b
otherwise
The characteristic function is given by
X ( w ) =
7.2
e
jwx
f X ( x ) dx =
b
jwx
jwx
e
e

Chapter 2
Random Variables
Section 2.4: Distribution Functions
2.1
We are given the following function that is potentially a CDF:
0
FX ( x ) =
( x 1 )
Bcfw_1 e
< x 1
1<x<
(a) For the function to be a valid CDF it must satisfy the condition
FX ( ) = 1 =
University of Massachusetts, Lowell
Department of Electrical and Computer Engineering
Course 16.363: Introduction to Probability and Random Processes
Second Midterm Exam, April 4, 2011
Name: gqm Fi
Instructions:
1. Answer all the questions in the spa
Solutions to 16.363 Problem Set 2, Spring 2011
L40 We are given a game that consists of two successive trials in which the rst trial has
outcome A or B and the second trial has outcome C or D. The probabilities of the four
possible outcomes of the game ar