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Discrete Random Variable CDF Lecture

This document provides a summary of a lecture on probability and statistics that covers the distribution function of a discrete random variable. It defines the distribution function as the sum of the probabilities of all values less than or equal to x. It then gives an example of finding the distribution function for the number of heads from tossing 3 coins. The document also provides properties of the distribution function and proves it is non-decreasing. Finally, it gives an example problem of calculating the distribution function, mean, variance, and cumulative distribution function for a discrete random variable.

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Sudip Bal Lama
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109 views7 pages

Discrete Random Variable CDF Lecture

This document provides a summary of a lecture on probability and statistics that covers the distribution function of a discrete random variable. It defines the distribution function as the sum of the probabilities of all values less than or equal to x. It then gives an example of finding the distribution function for the number of heads from tossing 3 coins. The document also provides properties of the distribution function and proves it is non-decreasing. Finally, it gives an example problem of calculating the distribution function, mean, variance, and cumulative distribution function for a discrete random variable.

Uploaded by

Sudip Bal Lama
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Lecture-9

ON
MATH-208

(Probability and Statistics)

BY
Kiran Kumar Shrestha

Department of Mathematics

School of Science

Kathmandu University

TO
CIVE - II – II (2019) Group Students

Topics Covered

 Distribution Function of Discrete RV

Date: Tue, 28 Jan. 2021


Distribution Function of a Discrete Random Variable
Example- Probability distribution of number of heads in tossing of 3 coins.

Here we see that ( ) is a function of 'x' as shown below-

( ) ( )
0 1/8
( ) ( )
1 3/8
( ) ( ) ( )
2 3/8
( ) ( ) ( ) ( )

3 1/8 ( ) ( ) ( ) ( ) ( )

Above result is expressed in following standard format-

( ) ( )

Defn.
Let X be a discrete random variable having probability mass function ( ), then its distribution function
is denoted as ( ) or simply ( ) and is defined as

( ) ( ) ∑ ( ) ∑ ( )

Notes:
#.1 It is also called cumulative distribution function (cdf).

#.2 ( )

#.3
( ) ( )

#.4

( ) ( )

#.5

( ) ( ) ( )
X<=b
Proof:
a< X <= b
X<= a

a b

From above diagram it is clear that the intervals are mutually exclusive and their
union is the interval X<=b, i.e.,

( ) ( ) ( )

So,

(( ) ( )) ( )

Or,

( ) ( ) ( )

So,

( ) ( ) ( ) ( ) ( )

#.6

( ) ( ) ( ) ( )

#.7

( ) ( ) ( ) ( )

#.8

( ) ( ) ( ) ( ) ( )

#.9 Distribution function is non-decreasing function of 'x'.

#.10

( ) ( ) ( )

Proof:

( ) ( ) ( ) ( )

* ( ) ( ) ( ) ( )+
* ( ) ( ) ( )+
( ) ( )

Problems:
#.1

Solution-

#.(a) We have,

∑ ( )

So,

( ) ( ) ( )

Or,

( ) ( ) ( )

Or,

Or,

#.(b) Mean is given by

( ̅) ∑ ( ) ( ) ( ) ( )

( ) ( ) ( )

#.(c) Variance is given by


∑ ( ) ̅ ( ) ( ) ( ) ( )

{ ( ) ( ) ( ) }

( )

#.(d) cumulative distribution function is given by

( ) ( ) ∑ ( ) ∑ ( )

When x = 1

( ) ∑ ( ) ∑ ( ) ∑ ( ) ( ) ( ) ( )

When x =2

( ) ∑ ( ) ∑ ( ) ( ) ( ) ( ) ( )

When x = 3

( ) ∑ ( ) ∑ ( ) ( ) ( ) ( )

( ) ( ) ( )

Thus, cdf is given by

( )

In standard format
( )

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