ECSE-2500 Engineering Probability HW#21 Solutions
Due 11/1/14
1. (8 points) Let X1 , X2 , X3 , X4 be random variables. Assume X1 , X2 are jointly Gaussian with
means 1, 2, respectively, variances 1, 4, respectively, and X1,X2 0.3. Assume X3 , X4 are joint

The Joint
Prob Density Function (PDF)
of Two Random Variables
Section 5.4
10/23/2014
ECSE-2500 Lecture 16: PDF of Two RVs
1
The Joint PDF of 2 RVs
Recall: Two random variables X and Y on a prob
space (S,P) have joint CDF defined by
FX ,Y x, y P X x,Y y

General Random Variables
& the Cumulative Distribution
Function (CDF)
Section 4.1
9/25/2014
ECSE-2500 Lecture 9: General RV & CDF
1
General Random Variables
So far, we have focused on Discrete RVs because
1. The math is simpler.
2. There are so many impo

Transform Methods:
Characteristic Functions &
Moment Generation
Section 4.7 except for 4.7.2-3.
Also, the last page of Sec. 4.6.
10/16/2014
ECSE-2500 Lecture 14b: Transform Methods
1
Transforms
In your Signals & Systems course, you learned about
the Four

Expected Value
& Continuous RVs
Sections 4.3 & 4.4
10/2/2014
ECSE-2500 Lecture 11a: RV Expected Value
1
Recall Expectation of a Discrete RV
The Expectation (aka Expected Value or Mean) of a
discrete random variable X was defined by
(1) mX E X
x p x
k

The Standard Gaussian,
the Q Function,
and Evaluating the
General Gaussian CDF
10/2/2014
ECSE-2500 Lecture 11b: Gaussian Q function
1
Gaussian RV
We have said (many times) that the Gaussian RV is
very important.
We have also noted that there is no close

The Markov and Chebyshev
Inequalities
Section 4.6
10/16/2014
ECSE-2500 Lecture 14a: Inequalities
1
Bounding Probabilities
Sometimes we cannot (or do not need to) compute
the CDF, but bounding it is sufficient.
Markovs Inequality: For any non-negative RV

CDFs & PDFs of a
Function of a RV
Y g X
Section 4.5
10/14/2014
ECSE-2500 Lecture 13: Functions of a RV
1
Expectation of a Function of a RV
Last time, we saw how to compute the expected
value of Y where Y is a function of a RV X, that is,
Y g X , via
E Y

Two Random Variables
Sections 5.13
10/20/2014
ECSE-2500 Lecture 15: Two RVs
1
Two Random Variables
A random variable represents a measurement on a
random experiment.
Many experiments require more than one
measurement.
When we model this with probabilit

Expectation of
a Function of Two RVs
and Joint Moments
Section 5.6
10/30/2014
ECSE-2500 Lecture 17: Joint Moments
1
Expectation of a Function of 2 RVs
Recall: If Y is a function of a random variable X,
i.e. Y = g(X), then
E Y
g x f
X
x dx
In a similar

Name: _Solutions_
RIN: _
Exam 2
Engineering Probability
ECSE-2500
November 6, 2014
Solutions
Points
1
20
2
20
3
25
4
35
Total
Actual Points
100
Notes:
Show all work.
You have 75 minutes to take this exam.
You are allowed one 8.5 x 11 inch piece of pape

ECSE-2500 Engineering Probability HW#4 Solutions
Due 9/11/14
1. (11 points) Let x be a point chosen at random in the real interval 0,2. Now let
A x 0,2 x 1 , B x 0,2 x 0.5 , and C x 0,2 x 1 0.25 .
1.a. (3 points) Calculate P A, P B, P C .
Solution
Uniform

The Probability
Density Function (PDF)
Section 4.2
9/29/2014
ECSE-2500 Lecture 10: Prob Density Function
1
Recall the CDF
The Cumulative Distribution Function (CDF) of a RV
X on a prob space (S,P) is denoted FX x and defined
for each x by
(2) FX x P X x