1.
ECE 361: Probability for Engineers: HW # 3
Test if the following expression is a valid probability density function
=
g ( x ) 3e x , x 0
Solution
g ( x )dx 1
No.
0
2.
Test if the following expression is a valid density function determine the constant
ECE 361: Probability for Engineers HW # 7
1. X is a random variable with the following pdf:
1
, 1< x < 9
8
f (=
x)
Determine the pdf of Y =
1
.
X
Solution: The transformation is a monotonic one,
f (=
y)
f ( x) 1
1 1
= ( 2 x 3/2=
, < y <1
)
dy
8
4 y3 3
dx
ECE 361: Probability for Engineers HW # 1 July 5 in class
1. The sample space of an experiment consists of the measured resistances of two resistors. Give
three examples of partitions.
2. A simple circuit is shown below. It has three subsystems.
(a) Expla
ECE 361: Probability for Engineers HW # 1
1. A thermometer measures temperatures from -40 to 130oF. (a) Define the universal set for this
measurement (b) specify the subset for temperature measurements not exceeding waters
freezing point (c) specify the s
The first 4 pages contain the solution prepared without the use of Matlab. The remaining pages contain the solution as a
Matlab published file containing the Matlab code and the results (all obtained using the symbolic toolbox).
p m shankar, April 16, 201
ECE 361: Probability for Engineers HW # 2
1. Two numbers x and Y are selected at random between 0 and 1. Three events are defined in
terms of the following outcomes: A=cfw_x>1/2; B=cfw_y>1/2 and C=cfw_x>y. Determine (a) if A and B
are independent (b) A an
1.
ECE 361: Probability for Engineers HW # 4 Due October 20
Find E[g(X)] ,
1 x
g ( X ) = X 2 where the function of X is f X ( x ) = e 2U ( x )
2
1 2x
E g ( X ) g (=
x ) f ( x ) dx =
x e dx 8
=
2
0
0
2
2. X is uniformly distributed [-5,15]. Find the mean o
ECE 361: Probability for Engineers HW # 2
1. Two numbers x and Y are selected at random between 0 and 1. Three events are defined in
terms of the following outcomes: A=cfw_x>1/2; B=cfw_y>1/2 and C=cfw_x>y. Determine (a) if A and B
are independent (b) A an
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 4b
2.6 Conditioning
Conditioning one RV on another
Example. A transmitter sends a message over a computer network. Define X as the travel time of the messag
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 6a
3.2 Cumulative distribution functions
Example. The maximum of several discrete uniform RVs. Let X = maxcfw_X1 , . . . , Xk where Xi Uni([n]) for i [k].
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 9b
4.2 Covariance and correlation
Example. (CONTINUED FROM PREVIOUS LECTURE).
Let the bins selected for the
n
nn balls be B1 , . . . , Bn , where each
Bt [k
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 4a
2.5 Joint PMFs of multiple RVs
Functions of multiple RVs
Just as we can discuss functions of a single RV Y = g(X), we can discuss functions of multiple R
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 7a
3.5 Conditioning
Conditioning an RV on an event
The total probability theorem for conditional PDFs states: given a collection of events A1 , . . . , An t
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 6b
3.3 Normal RVs
The Gaussian or normal RV X has support X = R and PDF
fX (x) =
(x)2
1
e 22 , x R.
2
(1)
We write X N (, ) to denote that X is an RV that
Instructors
ECE 361
Probability for Engineers
Fall 2017-2018
P. M. Shankar
Office:
Phone #: 215-895-6632
Bossone 311
Email: [email protected]
Hours: 9:30-11:00 AM (Tue/Thu)
Teaching Assistant(s)
Xinyi Wang
[email protected]
Hours: 4-6 PM (Tuesdays)
Main
Table of Contents
Random variables, distribution and density functions . 2
Differences between Discrete and Continuous random variables . 4
Properties
- Moments, CHF, MGF and Laplace transforms . 5
Examples of continuous and discrete random variables . 8
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 8b (preliminary)
4.1 Derived distributions
Functions of two RVs
To find the PDF of Z = g(X, Y ) we use the same two step procedure: i) find the CDF FZ , ii)
Lecture 7a
ECE 361
Probability for Engineers
Fall, 2016
Steven Weber
Tuesday November 3, 2016
Steven Weber (Drexel ECE)
ECE 361 Fall 2016 Lecture 7a
November 3, 2016
1 / 41
3.5 Conditioning
Outline
1 3.5 Conditioning
Conditioning an RV on an event
Conditi
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 7b
4.1 Derived distributions
Let the RV Y be a function Y = g(X) of a continuous RV X. We aim to calculate the PDF of Y , which we call the derived
distribu
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 5a
2.7 Independence
Independence of several RVs
We say RVs X, Y, Z are independent if their joint PMF factors as the product of the marginal PMFs
pX,Y,Z (x,
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 2b
Sample problems that use strategies similar to those required in HW 2
Problem. Fix events A, B in a sample space . Suppose you are given three probabilit
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 1a
1.1 Sets
A set S is an unordered collection of unique objects, called elements, with membership of element x in S denoted x S.
Empty (or null) set: .
Not
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 9a
4.2 Covariance and correlation
The covariance of two RVs, X, Y , is defined as:
cov(X, Y ) = E[(X E[X])(Y E[Y ])].
(1)
Say X, Y are uncorrelated when cov
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 2a
1.3 Conditional probability
Using conditional probability for modeling
Generalizing the arguments in the above example we identify the following multipli
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 3a
2.3 Functions of random variables
Functions of RVs. A function of a RV X, say Y = g(X), is a new RV.
Example. Let X be the temperature in degrees Celsius
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 3b
Some hints for homework 3
Hints on problems 1 and 8.
2.4 Expectation, mean, and variance
Mean and variance of some common RVs
Example. The mean and varia
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 10a
5.1 Markov and Chebychev inequalities
Example. Let X Uni[0, 4] with E[X] = 2. Then FX (x) = 1 FX (x) = P(X > x) = 1 x4 , E[X] = 2 and var(X) = 4/3.
The
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 10b
5.4 The central limit theorem
Given iid cfw_X1 , X2 , . . . with E[X] = < and var(X) = 2 < , we define cfw_M1 , M2 , . . . and cfw_S1 , S2 , . . . , as
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 8b
4.1 Derived distributions
Functions of two RVs
To find the PDF of Z = g(X, Y ) we use the same two step procedure: i) find the CDF FZ , ii) differentiate
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 1b
Some background for homework 1
Homework 1 problem 1 asks to find the probability of each outcome of the following experiment.
Consider flipping a fair co
Problem # 1:
ECE 361: Probability for Engineers HW # 5
Solution:
X is the current, the pdf of the current is
x2
x 2
f X ( x ) = 2 e 2b U ( x )
b
The power is
Y = X2
Since X is Rayleigh distributed, the power will be exponentially distributed as
y
1 2b2
e
ECE 361 HW # 2 Due July 11 in class
1. Five good fuses and two defectives ones are inadvertently mixed. To find the defective ones,
each one is tested one-byone. If the defective ones are identified only in the third attempt,
what is the probability of th
1. Solution:
ECE 361: Probability for Engineers HW # 8
August 22
f ( x, y=
) exp ( x y )U ( x )U ( y )
X
Z =X + Y W =
Y
This means that
=
x
z
x
J=
w
x
zw
x=
,
1+ w
z
y=
1+ w
zw
z
=
,y
1+ w
1+ w
z
1
y
=1
w
y
y
f ( z, w
=
)
1
2
1 + w)
(
x 1
x+ y
2 =
2 =
x
ECE 361: Probability for Engineers HW # 1 July 5 in class
1. The sample space of an experiment consists of the measured resistances of two resistors. Give
three examples of partitions.
Solution
Let R1 and R2 denote the measured resistances. The pair (R1,R