(10) hypergeometric distribution

7/30/2019 (10) Hypergeometric Distribution

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Applied Statistics and Computing Lab

HYPER-GEOMETRIC DISTRIBUTION


Indian School of Business


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LEARNING GOALS To study how Hyper-geometric distribution evolves

To learn its properties To understand various examples

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Probability distribution of X = number of units of type I in the sample of size n

Binomial distribution

(If selected with replacement)

Hyper-geometric distribution

(If selected without replacement)

Sample of n selected from the N objects or trials

Select with replacement Select without replacement

N objects of strictly 2 distinct types or N Bernoulli trials

M out of N of type I (N-M) out of N of type II

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Example Suppose 100 students are enrolled for a course in business analytics

Of these, 20 have a background in Statistics

Suppose we randomly pick 10 students from this cohort What is the probability that 3 out of these 10 would have a background in

Statistics?

There are

total ways of choosing 10 students from the entire class

How many ways are there to choose 3 students with background in Statistics, outof the 20 such students in the class?

Now, for each of these

combinations, there are

ways of choosing theremaining students of the sample, from the set of students who do not have abackground in Statistics

Hence, there are ways of choosing 3 students with Statistics backgroundand 7 without Statistics background, out of the total

ways

. =

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Formulating the PMF This example is nothing but a hyper-geometric distribution!

Notice that,

N=100 Type I objects or success of the trial is considered to be a student with Statistics background

M=20 and (N-M)=80

We are drawing a sample of size 10 n=10

We are interested in the probability that; of these 10, 3 have a background in Statistics X=3

Referring to our earlier formulation, we can state that

= =

where Max(0, n N M ) min( , ) and , , > 0

This is the probability distribution for a variable X following Hyper-geometricdistribution with parameters (N, M, n)

Denoted as ~ ( , , )

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Pictorial representation

M N-M

k n-k

N

n

Population

Sample

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Assumptions and statistics the PMF of Hyper-geometric distribution is defined under certain

assumptions:

The population is finite Population units are distinctly of two types

Sampling is done without replacement

For ~ ( , , ),

=

=()()

()

Important property:

As

The idea is, if the value of N is very large, it does not matter any further,

whether we replace the drawn sample or not7


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Further examples An official survey has recorded that only M out of the total N villages in

a particular district, have their own supply of potable drinking water. For

my field work, I am visiting n villages tomorrow, without any priorknowledge about the villages with and without drinking water. What arethe chances that I at least once during the day, I will get to drink waterfrom the village I am visiting?

In a shipment of 1000 computer monitors, 7 are defective. Suppose amanager checks 50 monitors, every time a new shipment of 1000 is to besent out. What is the chance that 10 of these would be defective?

Suppose the India office of a global consulting firm accepts N projects

every year, out of which M are based out of a foreign country. Everyemployee at the senior manager level works on n projects during anyparticular year. Then, what is the probability that one of the recentlypromoted senior managers would get a chance to work on kinternational projects next year?

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Thank you

(10) hypergeometric distribution

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