Chapter 6
Expectation and Conditional Expectation
Lectures 24 - 30
In this chapter, we introduce expected value or the mean of a random variable. First we define expectation for
discrete random variables and then for general random variable. Finally we in
Assignment 1
1. Let
be a finite set and
is a field of subsets of
. Show that
is a
-field.
2. Write down the expression in set notation corresponding to each of the following events.
(i) The event which occurs if exactly one of the events
(ii) The event wh
Assignment 9
1. Let
and
be independent geometric random variables with parameters
respectively. Find
2. Let
parameter
3. Let
and
and
.
be independent and identically distributed Poisson random variables with
. Find
.
be a nonnegative integer valued random
Chapter 3
Conditional Probability and Independence
Exercises
3.1 Show that constant random variable is independent of any other random variable defined on the
same probability space.
3.2 For
is a
, show that
-field.
3.3 Prove Theorem 2.0.8 .
3.4 Consider
Chapter 5
Random vectors, joined distributions
Exercises
5.1 Show that
for
.
5.2 Show that
contains countable union of closed sets of
.
5.3 Show that
5.4 Let
and
be independent random variables with uniform distribution on
random variable
is with uniform
Chapter 6
Expectation and Conditional Expectation
Exercises
6.1 Let
6.2 Let
a.s. Show that
and
.
be two nonnegative discrete random variables such that
. Show that
.
6.3 Let
be a non negative continuous random variable with pdf
6.4 Let
be a continuous ran
Chapter 1
Introduction
Exercises
1.1 Let
Show that
is a probability space such that
is a finite set.
1.2 Let a point is picked at random from the unit square
it is in the triangle bounded by
1.3 Let
Find
1.4 A box has
and
be a probability space. Let
and
S
Chapter 7
Characteristic functions
Exercises
7.1 Let
. Show that the characteristic function of
Here
7.2 Let
means
and
is normally distributed with mean
is given by
and variance
.
be independent random variables. Show that
7.3 Let
. Find the characteristi
Chapter 8
Limit Theorems
Exercises
8.1 Let
be a random variable taking values in
such that
. Show that
.
8.2 Let
be a sequence of independent random variables such that
Show that
8.3 Let
doesn't converge to 0 a.s.
in probability. Is it true that
in probab
SCHOOL OF ENGINEERING & BUILT
ENVIRONMENT
Mathematics
Probability and Probability Distributions
1.
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5.
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20.
Introduction
Probability
Basic rules of probability
Complementary events
Addition Law fo
Section 2.3
1
Combinatorics
Section 2.3
2.3 Combinatorics
Purpose of Section To introduce some basic tools of counting, such as the
multiplication principle, permutations and combinations.
Introduction
If someone asks you a question that starts off how ma
EE 325: Probability and Random Processes
Instructor: Saravanan Vijayakumaran
Indian Institute of Technology Bombay
Spring 2013
Selected Solutions to Assignment 1
1. For a sample space , prove the following statements.
(a) If A , F = cfw_, A, Ac , is a -f
Assignment 7
1. Let
and
independent Poisson random variables with parameters
respectively. Find
.
2. Let
and
be independent random variables each having exponential density with parameter
. Find the conditional pdf of
3. Let
and
given
.
be discrete random
Assignment 6
1.
Let
and
be
independent
identically
distributed
. Find (i)
2.
Let
and
be
independent
random
variables
(ii)
identically
.
distributed
random
variables
. Find the distribution function of
3. Let
and
be continuous random variables with joint p
Assignment 2
1. Show that any countable union of closed sets in
2. Let
be a probability space. Show that
3. Let
is a random variable iff
be a probability space. Then
for all open set
4.Let
is a Borel set.
and
of
.
is a random variable iff
.
be random vari
Assignment 3
1. Let
be a probability space. Let
be independent and satisfies
. Show that
2. Let
be a probability space and
be in
such that
Show that
3. Let
be a probability space. Let
. Then
4. Let
and
6. Let
is independent of
iff
be a probability space a
Assignment 5
1. Let
be a random variable with distribution function
. Show that
2. Give an example of a random variable which is neither discrete nor continuous.
3. Define
Show that
4.
Let
by
is a distribution function. Does pdf exists?
be
a
random
variab
Assignment 4
1. Give an example of
such that
2. Let
be sequences of subsets of a non empty set
. Show that
3. Let
be sequences of subsets of a non empty set
. Show that
4. Give an example of sequences of sets
5. Let
6. Let
such that
be a sequence of subse
Assignment 10
1. Let
be a sequence of random variables and
distribution iff
2. Let
3. Let
. Then
in
in probability.
in probability. Show that
in probability.
be a sequence of random variables and taking values in
variable taking values in
such that
for ea
Assignment 8
1. Let
be a continuous random variable with pdf
such that
. Without
using Theorem 6.0.32, show that
2. Let
be a continuous random variable with pdf
and
is an increasing (strictly) differentiable
function with derivative continuous. Without us
Chapter 3
Conditional Probability and Independence
Lectures 8 -12
In this chapter, we introduce the concepts of conditional probability and independence.
Suppose we know that an event
already occurred. Then a natural question asked is `What is the effect
Chapter 2
Random Variables
Lectures 4 - 7
In many situations, one is interested in only some aspects of the random experiment. For example,
consider the experiment of tossing
unbiased coins and we are only interested in number of 'Heads'
turned up. The pr
Chapter 1
Introduction
Lectures 1 - 3.
In this chapter we introduce the basic notions random experiment, sample space, events and probability
of event.
By a random experiment, we mean an experiment which has multiple outcomes and one don't know in
advance
Chapter 7
Characteristic functions
Lectures 31 - 33
In this chapter, we introduce the notion of characteristic function of a random variable and study its
properties. Characteristic function serves as an important tool for analyzing random phenomenon.
Def
Chapter 8
Limit Theorems
Lectures 35 -40
Most important limit theorems in probability are `law of large numbers' and 'central limit theorems`.
Law of large numbers describes the asymptotic behavior of the averages
, where
is a sequence of random variables
Chapter 4
Distributions
Lectures 13 - 17
In this chapter, we associate with random variable, a nondecreasing function called distribution function
and study its properties.
Definition 4.1. (Distribution function) Let
. The function
be
defined by
is called
Chapter 5
Random vectors, Joint distributions
Lectures 18 -23
In many real life problems, one often encounter multiple random objects. For example, if one is
interested in the future price of two different stocks in a stock market. Since the price of one
Chapter 2
Random Variables
Exercises
2.1 Let
be a constant function. Show that
2.2 Let
be a constant function. Find
is a random variable.
.
2.3 Give an example of a function on a probability space which is not a random variable.
2.4 Let
be a random variab