Weekly Notes for EE2012 2014/15 Week 3
T. J. Lim
February 2, 2015
Book sections covered this week: 2.4 2.6.
1
Conditional Probability
1.1
Denition
A central application of probability theory is to answer questions of the form If
event B occurs, how does t
EE2012 2014/15 Problem Set 5
Conditional PMF and Expected Values
1. Let A = cfw_X 3. Find the conditional PMF pX (x|A) for the following random
variables:
(a) X is binomial with n = 5, p = 0.2.
5
Ans: We have pX (k) = k 0.2k 0.85k , k = 0, 1, . . . , 5. T
EE2012 2014/15 Problem Set 11
Conditioning on a Random Variable
1. Let X be a continuous uniform random variable in [1, 1], and suppose the conditional PDF of Y given X is
fY |X (y|x) = |x|e|x|y ,
y > 0.
(a) Find P [Y > X] by rst nding P [Y > X|X = x] and
EE2012 2014/15 Problem Set 4
Discrete Random Variables
1. An urn contains nine $2 notes and one $10 note. Let the random variable X
be the total amount that results when two bills are drawn from the urn without
replacement.
(a) Describe the underlying sam
EE2012 2014/15 Problem Set 2
Probability Laws
1. Pick a real value in the range [0, 1]. Let
A = [0, 0.5];
B = (0.4, 0.8];
C = (0.6, 1.0].
(a) If the distribution function is
0 x<0
x2 0 x < 1
F1 (x) =
1 otherwise
nd P [A B], P [B C c ] and P [Ac B C].
EE2012 2014/15 Problem Set 4
Discrete Random Variables
1. An urn contains nine $2 notes and one $10 note. Let the random variable X
be the total amount that results when two bills are drawn from the urn without
replacement.
(a) Describe the underlying sam
EE2012 2014/15 Problem Set 3
Conditional Probability
1. A die is tossed twice and the number of dots on the top face noted in the order of
occurrence. Let A = rst toss second toss, and B = rst toss is a 6. Find
P [A|B] and P [B|A].
Ans: It should be obvio
EE2012 2014/15 Problem Set 3
Conditional Probability
1. A die is tossed twice and the number of dots on the top face noted in the order of
occurrence. Let A = rst toss second toss, and B = rst toss is a 6. Find
P [A|B] and P [B|A].
2. A number x is select
EE2012 2014/15 Problem Set 1
Set Theory
1. For each of the following sets, write down two subsets:
(a) R, the set of real numbers.
(b) C, the set of complex numbers.
(c) Z, the set of integers.
(d) S = [0, 2), the set of all real numbers from 0 to 2, incl
EE2012 2014/15 Problem Set 5
Conditional PMF and Expected Values
1. Let A = cfw_X 3. Find the conditional PMF pX (x|A) for the following random
variables:
(a) X is binomial with n = 5, p = 0.2.
(b) X is geometric with p = 0.4.
(c) X is Poisson with E[X] =
EE2012 2014/15 Problem Set 2
Probability Laws
1. Pick a real value in the range [0, 1]. Let
A = [0, 0.5];
B = (0.4, 0.8];
C = (0.6, 1.0].
(a) If the distribution function is
0 x<0
x2 0 x < 1
F1 (x) =
1 otherwise
nd P [A B], P [B C c ] and P [Ac B C].
Weekly Notes for EE2012 2014/15 Week 6
T. J. Lim
February 18, 2015
Book sections covered this week: 3.5, 4.1.
1
Important Discrete Random Variables
Discrete r.v.s often represent a count of the number of occurrences of some random
phenomenon, and certain
EE2012 2014/15 Problem Set 1
Set Theory
1. For each of the following sets, write down two subsets:
(a) R, the set of real numbers.
Ans: (0, 1), cfw_3, 4, 2, etc.
(b) C, the set of complex numbers.
Ans: R, jR, etc.
(c) Z, the set of integers.
Ans: Natural
Weekly Notes for EE2012 2014/15 Week 5
T. J. Lim
February 6, 2015
Book sections covered this week: 3.4
1
Conditional Distributions
We previously introduced the concept of the conditional probability of an event A
given the occurrence of another event B, d
Weekly Notes for EE2012 2014/15 Week 7
T. J. Lim
March 5, 2015
Book sections covered this week: 4.1 4.2.1.
1
Cumulative Distribution Function (CDF) (contd)
1.1
Types of Random Variables
The classication of a r.v. into one of the three categories of random
Weekly Notes for EE2012 2014/15 Week 9
T. J. Lim
March 19, 2015
Book sections covered this week: 4.5, 5.1
1
Functions of a Random Variable
1.1
Problem Statement
Random variables appear in engineering problems in many guises, e.g. as a time
interval, as a
Weekly Notes for EE2012 2014/15 Week 11
T. J. Lim
March 26, 2015
Book sections covered this week: 5.5, 5.6.
1
1.1
Independence of X and Y
Concept
Recall that we had earlier introduced the concept of independence between two
events A and B:
A, B independen
Weekly Notes for EE2012 2014/15 Week 10
T. J. Lim
March 19, 2015
Book sections covered this week: 5.25.4.
1
Both X and Y are Discrete
1.1
Joint Probability Mass Function (PMF)
When X and Y are both discrete (as in Example B), then we would normally use
th
Weekly Notes for EE2012 2014/15 Week 12
T. J. Lim
April 9, 2015
Book sections covered this week: 5.7.
1
Conditioning on a Random Variable
1.1
Concept
We have encountered conditional probabilities of the form P [A|B] where A and
B are both events, and we h
Weekly Notes for EE2012 2014/15 Week 13
T. J. Lim
April 14, 2015
Book sections covered this week: 5.8.1.
1
Functions of Two Random Variables
When a random variable Z = g(X, Y ), there are several ways of obtaining its
distribution, depending on whether it
Weekly Notes for EE2012 2013/14 Week 8
T. J. Lim
March 10, 2015
Book sections covered this week: 4.3 4.4.
1
Expected Values
1.1
Denition
For a discrete random variable X, the expected value was dened in an earlier class
as
E[X] =
xk pX (xk )
k
where pX (x
EE2012 2014/15 Problem Set 6
CDF and PDF
1. A random variable X has the PDF
1
fX (t) = et/4 ,
4
t > 0.
(a) Find the CDF of X.
Ans: By denition, we have
x
FX (x) =
fX (t)dt
(1)
x
=
0
1 t/4
e
dt,
4
et/4
=
= 1e
x0
(2)
0
x
x/4
(3)
,
x 0.
(4)
(b) Using the CDF
EE2012 2014/15 Problem Set 7
Expected Values
1. Find the mean and variance of the following PDFs:
(a) fX (x) = 5 (1 x4 ), 1 < x 1.
8
(b) fY (y) = 6y(1 y), 0 y 1.
(c) fZ (z) = 0.5(z + 2) + 0.25[u(z) u(z 2)].
2. Let the function g(x) be dened as
a x a
x a
EE2012 2014/15 Problem Set 6
CDF and PDF
1. A random variable X has the PDF
1
fX (t) = et/4 ,
4
t > 0.
(a) Find the CDF of X.
(b) Using the CDF, nd P [1 < X 2] and P [X > 4].
(c) X is a continuous random variable. Can you think of a function g(x) that
wil
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