Download Chapter 4 Distributions Lectures 13 - 17 Definition 4.1. (Distribution

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 the distribution function of
Theorem 4.0.13
The distribution function has the following properties
(i)
(ii)
is non decreasing.
(iii)
is right continuous.
Proof:
(i) Let
, set
Then
Therefore
i.e.,
Hence
Using similar argument, we can prove
(ii) For
. Hence
(iii) We have to show that, for each
Let
. Set
.
a
random
variable on
Then
and
Therefore
Theorem 4.0.14
Proof. Let
The set discontinuity points of a distribution function
be the distribution function of a random variable
is countable.
and
Then
where
Set
Then
Also note that for
If
be distinct points in
. Then
Therefore
Hence
is countable.
Theorem 4.0.15 Let
define
be a random variable on a probability space
. For
,
Then
is a probability measure on
.
Proof.
Also
for all
. Let
[note that
be pairwise disjoint, then
's are disjoint imply that
's are disjoint]
This completes the proof.
Definition 4.2 The probability measure
and we denote it by
Note that if
is called the distribution or law of the random variable
.
is the distribution function of
, then
Example 4.0.1 Consider the probability space
Define
as follows. For
= number of heads in
Then
takes values from
given by
,
.
. The distribution function of
is
The distribution function of the random variable is given by
Example 4.0.2 (Bernoulli distribution) On
Then
is a probability measure on
, for
, define
called the Bernoulli distribution.
A random variable with Bernoulli distribution is called a Bernoulli
random variable.
For example: Let
and define the probability measure
Define
on
such that
by
Then
Hence
is a Bernoulli random variable. The distribution function corresponding to Bernoulli
distribution is given by
Example 4.0.3 (Binomial distribution with parameters
given by
).On
,the probability measure
Example 4.0.29
Example 4.0.30
Example 4.0.31
Example 4.0.32
Classification of Random Variables. Random variables can be classified using Distribution
function
Defination 4.3:
Defination 4.4:
Defination 4.5:
Defination 4.5:
A continuous random variable with pdf is said to be an absolutely continuous random variable.
Example 4.0.33: