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
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
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
MANUFACTURING PROCESSES II
INJECTION MOLD HOMEWORK ASSIGNMENT
04/23/02
1. Determine the values of the items listed below which are required to successfully mold 8 plastic enclosures. The
dimensions of the enclosure are provided on page 3 and dimensions fo
Problem#1
a. Three events A, B, and C have probabilities respectively of P(A), P(B) and P(C). Furthermore,
C B A. What is the probability of the event A B C?
Answer: P(C)
b. X is a picked randomly in the interval [0,1]. What is the probability of the even
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Midterm Exam Solution
1. 4 points. Presents. Fix integers k, n with 0 < k < n. A mischevious parent places k toys in n (closed)
boxes and wraps up each box as a pre
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
Steven Weber
Dept. of ECE
Drexel University
ECE 361 Probability for Engineers (Fall, 2016)
Lecture 5b
3.1 Continuous RVs and their PDFs
Expectation
The expectation of a continuous RV is:
E[X] =
xfX (x)dx
(1)
This definition should be expected given the co
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 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 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 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 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
Solution to Problem 1.20. Figure 1.1 provides a sequential description for the
three different strategies. Here we assume 1 point for a win, 0 for a. loss= and 1/2 point
Bold play
(a) (b)
Bold play
Bold play for a draw. In the case of a tied 1—1 score
2.4 Expectation & Variance
Recitation
Homework
2.5-2.6 PDF & CDF
Recitation
A ball is drawn from an urn containing 3 white and 3 black balls. After the ball is
drawn, it is then replaced and another ball is drawn. This goes on indefinitely.
What is the pr
Chapter 2 Discrete Random
Variables
1
2.1 What is a Discrete Random
Variable?
Consider an experiment that can have many outcomes
Let X represent all of the possible outcomes of the
experiment
Map all possible outcomes on the real number line
Each possibl
1.
2.
3.
ECE 361: Probability for Engineers HW # 3, due April 21
Test if the following expression is a valid probability density function
= 3e x , x 0
g ( x)
Test if the following expression is a valid density function and if it is valid, determine the
co
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
1.
ECE 361: Probability for Engineers HW # 4, due April 28
Find E[g(X)] with
g(X ) = X2
The density function of X is
1 x
f X ( x ) = e 2U ( x )
2
2. X is uniformly distributed [-5,15]. Find the mean of Y = e
3.
X
5
For the following density function,
?
x
ECE 361: Probability for Engineers HW # 6, due May 12
1. The voltage X generated by a noise source is Gaussian N(0,9). This is passed through a square
law device with the output Y=5X2. (1) What is the density function of the output of the square
law devic
1.
ECE 361: Probability for Engineers HW # 5
A current having a Rayleigh density is passed through a resistor of Ohms. The current has an
average value of 3 Amps. (a) Using the definition of the pdf and CDF, obtain an expression for
the pdf of the power d
ECE 361: Probability for Engineers HW # 6 due 16
1. The voltage X generated by a noise source is Gaussian N(m,s2). This is passed through a square
law device with the output Y=10X2. (1) What is the density function of the output of the square
law device (
1.
ECE 361: Probability for Engineers HW # 5
A current having a Rayleigh density is passed through a resistor of Ohms. The current has an
average value of 3 Amps. (a) Using the definition of the pdf and CDF, obtain an expression for
the pdf of the power d
1.
Find E[g(X)] ,
ECE 361: Probability for Engineers HW # 4
1 x
g ( X ) = X 2 where the function of X is f X ( x ) = e 2U ( x )
2
x
1
=
E g ( X ) g (=
x ) f ( x ) dx =
x 2 e 2 dx 8
2
0
0
2. X is uniformly distributed [-5,15]. Find the mean of Y = e
=
E [
ECE 361: Probability for Engineers HW # 6 due May 16
1. The voltage X generated by a noise source is Gaussian N(m,s2). This is passed through a square
law device with the output Y=10X2. (1) What is the density function of the output of the square
law devi