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8/11/2019 05 Discrete Probability Distribution

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Random Variables &

Discrete Probability Distribution

Dr. Sohail Iqbal

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Math-801 Mathematical Methods for Computing


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Outline

Concept of Random Variable

Examples of Random Variable Problems

Discrete Probability Distribution The Binomial Distribution

Examples

Home Work

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Concept of

Random Variable

In most probability problems, we are

interested in one number that is associated

with the outcome of experiment. Such

number, being random due to random

outcome of experiment, is called Random

Variable.

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Examples of

Random Variable Problems

1. In manufacturing,x= No. of defective items

2. In Road Testing, y= Average vehicles speed

3. Throwing two dice, z = sum of two dice

All these numbers are associated with

situations involving randomness, therefore,all of the above variable x, y, and z are three

different random variables.

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Probability Distribution

Studying Random Variables we interest in:

Probabilities with which they take the various

values in their range

This spread of probabilities for the various

values of random variables is called a

probability distribution

In the following, we construct an example:

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Experiment of

Throwing two Dice

Results:

Tabulated:

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Think about having a

sum of two dice= z =7

P(X=7)= 6/36


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Example 1

Probability distribution function (pdf) for the sum

of two dice

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Note: Above diagram is experimental, make your self ideal pdf.


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Example 2

Note: Probability

distribution

function is also

called probability

density function

or probabilitymass function.

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Discrete Probability Distribution

When the random variable takes the discrete

values, its probability distribution will be

Discrete Probability Distribution.

For example, throwing two dice or number of

defective items in each shipment will have

discrete probability distributions.

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Pattern Understanding

By understanding the underlying pattern

behind each experiment, we have formulated

different probability distribution.

In the following, we shall study the most

important discrete probability distribution

called The Binomial Probability Distribution.

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Binomial Probability Distribution

Consider experiment of tossing a coin:

1. Outcome can be classified as success/failure

2. Success probability,p, remains same3. The successive trials are all independent

Any experiment where trials respect above

three conditions are called Bernoulli trails.The experiment having n Bernoulli trials is

called a binomial probability experiment.

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Binomial Probability Distribution

WhenXdenotes the number of successes in n

trials of a binomial probability experiment, it

is called a binomial random variable having

Binomial Probability Distribution. For such r.v.

X, the binomial p.d. is given by the formula:

where q =1-p is the probability of failure12

( ; , ) ( ) , 0,1,2,...,x n xn

b x n p P X x p q x n x


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Classic Example

Q: A coin is tossed 5 times. What are probabilities

of obtaining various number of heads?

Since these are Bernoulli trials with n=5 times.

The r.v.Xhas a binomial probability distribution

with p=1/2, q=1/2, and n=5. Therefore we apply

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( ; , ) ( ) , 0,1,2,...,x n xn



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( ; , ) ( ) , 0,1,2,...,x n xn


Main formula for BPD

551 1 1( ; 5 , ) ( ) , 0,1,2, 3, 4, 5

2 2 2

x x

b x P X x x x

0 5 0

51 1 1 1 1(0; 5 , ) ( 0) 1 102 2 2 32 32

b P X

3 5 351 1 1 1 1 10

(3; 5 , ) ( 3) 1032 2 2 8 4 32b P X


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Reading for Quiz 1

http://en.wikipedia.org/wiki/Probability

http://en.wikipedia.org/wiki/Conditional_pro

bability

http://en.wikipedia.org/wiki/Bayes'_theorem

With all, we have done and assignment 1.

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http://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Conditional_probabilityhttp://en.wikipedia.org/wiki/Conditional_probabilityhttp://en.wikipedia.org/wiki/Conditional_probabilityhttp://en.wikipedia.org/wiki/Conditional_probabilityhttp://en.wikipedia.org/wiki/Bayes'_theoremhttp://en.wikipedia.org/wiki/Bayes'_theoremhttp://en.wikipedia.org/wiki/Bayes'_theoremhttp://en.wikipedia.org/wiki/Conditional_probabilityhttp://en.wikipedia.org/wiki/Probability


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Questions?

Thank You!

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