05 discrete probability distribution
TRANSCRIPT
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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
b x n p P X x p q x n x
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( ; , ) ( ) , 0,1,2,...,x n xn
b x n p P X x p q x n x
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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