Download Poisson Distribution

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A
Presentation on
DiFFERENCES B/W
DiSTRiBUTiONS
Brief introduction of Binomial, Hypergeometric,
Poisson, Normal Distributions.
Presented by
NAIMA SABIR………..Roll # 909
ASMA SAJID……………Roll # 936
BCS-II
Presented before
Sir. YAHYA SAEED
COMPUTER SCIENCE DEPARTMENT
GCUF
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Objective
To provide easy understandability of all distributions so that one may easily
judge where to apply which distribution. Differentiating b/w these
distributions helps to apply the exact one.
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OUTLINE
 Binomial Distribution.
 Normal/Gaussian Distribution.
 Poisson Distribution.
 Hypergeomtric Distribution.
Definition
Mathematically
Why we use it
Properties
Daily life example
Numerical
 Differences b/w all.
 Application of Hypergeometric
Distribution(proceedings of annual Meeting of American
Statistical Association Aug5-9,2001)
 Summary.
 Question & Answering Session.
---------------------------
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Binomial Distribution
HISTORY: The first one distribution, which was developed, was Binomial Distribution.
Although there is some other statements of satiations, which tells that hypergeometric
was the first one to be made but it, doesn’t seems to be true.
Binomial experiment is that experiment in which out comes are always classified in
success or in failure.
Type:- This is distribution of Discrete type
Definition: “A distribution based on Binomial experiment is called Binomial Distribution.”
OR
“The probability distribution of a numerical quantity whose value is determined by
the implementation of Binomial experiment is called Binomial Distribution.”
Mathematically: Its mathematical representation is given by;
If “p” is probability of success & “q” is probability of failure then from n samples
Binomial distribution is
P (x; n, p, q) =nCx px qn-x
Here x =0,1,2,3,4,5,. . .
Why we use it?: When our possible outcomes are only two. when we have success or failure
It describes the possible no. of times that a particular event will occur in a sequence of
observations. The event is coded binary; it may or may not occur.
Properties: 1.
2.
3.
4.
5.
6.
The probability of success remains constant.
Successive trails are independent.
Experiment is repeated for a Large no. of times.
Each trail outcome has two categories; success or fail.
Its general form depends upon parameters p and q.
It may also be applied to a finite population if it is not very small.
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Features of binomial distribution:µ=np
σ2=npq
σ=√npq
Mean
Variance
Standard Deviation
Daily life implementation:  A coin tossed during a cricket match. What is the probability that Pakistani
captain gets head.
 When a researcher is interested in the occurrence of an event not in its magnitude.
 A baseball player’s batting average is 0.250.what is probability that he gets
exactly one hit in his next 5 times at bat?
Numerical: What is the probability of rolling two sixes & three nonsixes in 5
independent cast of a fair die?
Solution
Here,
n=5
X=2
p+q =1
P=1/6 so
q=1-(1/6)=5/6
Putting values in it
P (2)=5C2 (1/6) 2 (5/6) 3
Numerical 2: A baseball player’s batting average is 0.250.what is probability that he gets
exactly one hit in his next 5 times at bat?
Solution
Here
p=0.25
q=1-p ,q=0.75
n=5 ,x=1
Using these values we get,
P (1)=5C1 (0.25) 1(0.75)
P=5 x 0.25(0.75)
P=0.39550 Ans
5-1
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important
 Binomial approaches Poisson for large “n” and small”
 Binomial approaches Normal if no. of observations are
large
Hypergeometric Distribution
Hypergeometric Experiment:An experiment in which a random sample is selected from a “known finite”
population “without replacement”.
Definition:“The probability distribution of hypergeometric experiment is called
Hypergeometric Distribution.”
OR
“In general we are interested in the probability of selecting “x” successes from “k”
items labeled “success”& “n-x” items labeled as failures. When a random sample of
size “n” is selected from a finite population of size “N”.”
Mathematically: It is denoted by h and ids given by,
h(x; k, n,N)=kCx
(N-k)C(n-x)/NCn
Why we use it?
When there are two events and we are interested in one of it.
For example there are men and women in a particular firm and our interest is in men of
that firm. Here we ‘ll use hypergeometric distribution.
Properties: 1)
2)
3)
4)
5)
Each trail is classified into success & failure in one of them we are concerned.
Successive trails are dependant
The probability of success will change after each trail.
Experiment is repeated for a fixed no of time.
If we set k/N=p, then the mean of hypergeometric distribution coincides with
mean of binomial distribution.
6) When N is large hypergeometric approaches the “Binomial distribution”.
Daily life implementation: For lotto games like “PowerBall” and “Gopher 5”binomial distribution doesn’t apply.
Because in lotto games events are not independent. These are events without
replacement. The selection of successive balls is not put back into the machine.
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For “Power ball “ probability of first ball being a 1 is 1/53.if a 1 ball is selected,
the probability of a 1 ball on second ball gets Zero. If 1 ball is not selected on first
attempt, its selecting probability drops to 1/52 because there is one ball lesser in the
machine.
This is referred to as “sampling without replacement”.
Numerical: What will be the probability of matching three of five white balls in
“Powerball”game?
Solution
Here we have
Total balls =N=53
No. of white balls =k=5
No. of balls that match a no. on our ticket =n=5
No. of balls drawn = x =3
So applying H.d
H(3;5,5,53) =(5C3) ( 48C2
H(3;5,5,53) =10 x1128/2869685
=0.003931 Ans.
)
/ (53C5 )
Using Ms Excel
A built-in function is used to calculate hypergeometric distribution called
HYPEGEMDIST
……………………………….
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Poisson Distribution
Type:-This is a discrete type Distribution.
Definition: “It is defined as the limiting case of Binomial distribution when “n” is very large & “p” is
very small.”
Mathematically: Its probability distribution is given by
P(x=( e- µ µx ) /x!
Why we use it?
It is used when the probability of success is very very small.
When the experiment in which we are concerned has very less success probability.
Properties: 1)
2)
3)
4)
5)
6)
7)
The probability of success is very small.
Trails are independent
In Poisson Mean & variance are equal MEAN=VARIANCE= µ
Average is involved.
Experiment is repeated for a fixed no. of times.
For small values of meu, it is not symmetrical but skewed.
It resembles the Binomial if the probability of an event is very small.
Daily life implementation: 



It is widely used in biology.
It used to find no. of bacteria on a plate
It is used to calculate no. of miss prints in a 1000 or more page book.
It was firstly defined for the description of no of deaths by hoarse kicking in
Prussian army.(By Poisson in 1837).
 It is applied to the probability of mutations in a DNA Segment.
 It is also used to describe the distributions of counts detected from a Radioactive
sample.
Numerical: A secretary makes 2 errors per page on average. What is the probability that
on next page she makes no errors ?
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Solution
Here
so
x=0.
µ=2
P(0;2)=e-2 20/0!
p(0;2)=0.13536 Ans
NORMAL DISTRIBUTION
Type:- it is a continuous type distribution
History: - In 1733 Demoivre derived the mathematical equation of
“Normal curve”. It is often referred to as “Gaussian Distribution” in
honor of “Gauss (1777-1855).
Definition: “It is a limiting case of Binomial Distribution when “n” is very large & “p” is
moderate.
Mathematically: if x is a normal random variable with mean µ & variance σ then equation of the normal
curve is
n(x; µ, σ) =e-1/2(x-µ/ σ )2/ σ√2∏
-∞< x <+∞
Why we use it?
It is generally used when we are given σ or µ to find the area under the curve when we
are given to find the specific area under the curve. It is always converted in to Standard
Normal Distribution for find in area under the curve.
Properties: 1)
2)
3)
4)
5)
It is symmetrical if MEAN=MEDIAN =MODE
If µ=0 and σ=1 we convert Normal in to Standard normal
If n (a1<x<a2) then Ф (a2)-Ф (a1)
If n (a>x) then 1- Ф (a)
It is a distribution of infinite variables.
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Differences B/w Distributions
In this section we ‘ll discuss differences among these distributions
Binomial
Distribution
Hypergeom
etric
Distribution
Poisson
Distributio
n
Normal
Distribution
“The
probability
distribution of
hypergeometric
experiment is
called
Hypergeometri
c Distribution.”
It is defined as
the limiting case
of Binomial
distribution when
“n” is very large
& “p” is very
small.”
“It is a limiting
case of Binomial
Distribution
when “n” is very
large & “p” is
moderate
P(x=( e- µ µx ) /x!
n(x; µ, σ) =e-1/2(x-µ/ σ
)2/ σ√2∏
When we r
given sigma or
meu
Definition
“A distribution based
on Binomial
experiment is called
Binomial
Distribution.”
Notation
P (x; n, p, q) =nCx px qn-x
h(x; k, n,N)=kCx
k)C(n-x)/NCn
When needed
When possible
outcomes are 2
Parameters
Value of u
Mean
comparison
Probability
P,n
Not exist
Irrelevant
We have two or
more events and
we r interested in
one
N, k, n
Not exist
Irrelevant
When
probability of
success is very
small
µ
Probability involves Probability
involves
With Replacement
Without
replacement
Trails are
Trails are
independent
independent
Experiment is
Experiment is
repeated for a fixed repeated for a
no. of time
fixed no. of time
Average
involves
Not concerned
Not concerned
Independent
trial
Experiment
repeated for a
fixed no. of
time
Not related
Dependent
variable exists
Does not exist
Replacement
Trails
No. of
experiments
Constant
Probability of
success remains
constant
Not remains
constant
(N-
µ=np
Mean
=variance= µ
µ, σ
Any given
Mean=median=
mode
Not concerned
Not related