a4 discrete probability models

7/27/2019 a4 Discrete Probability Models

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Discrete Probability Models

Arun Kumar, Ravindra Gokhale, and NagarajanKrishnamurthy

Quantitative Techniques-I, Term I, 2012

Indian Institute of Management Indore
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Probability Models

Standard probability models (probability distributions) areavailable in the literature and have been studied in detail.These models can mimic many real life scenarios very well andhave mathematically tractable representation.
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Discrete Probability Distributions

Binomial Distribution*

Poisson Distribution*

Hypergeometric Distribution

Geometric Distribution

Multinomial Distribution
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Bernoulli Experiment

An experiment that results in only two outcomes.
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Example

1 Toss of a coin.

2

Choice of voters (Democratic candidate or Republicancandidate).

3 An item is defective or not.

4 Pass or fail.

5 Have a disease when you have certain symptoms or doesnot have the disease.
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Binomial Experiment

*Repeated Bernoulli experiments.
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Binomial Experiment

n identical Bernoulli trials.

Label one of the outcomes as success and other as failure.

P(success)=p and P(failure)=1 p. p and 1 p are thesame for all trials.

Trials are independent.
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Exercise 1

A jar contains five balls: three red and two white. Two ballsare randomly selected without replacement from the jar, and

the number of x red balls are recorded. Explain why x is or isnot a binomial random variable?
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Binomial Distribution

Probability ofk success in n trial is

P(X = k) =

n

k

pk (1 p)nk

where k {

0, 1, 2, 3, . . . , n}

.
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n

kand Factorial
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Understanding the Distribution Formula

Probability ofk successes is pk.Probability ofn k failures is (1 p)nk.k successes in n trials can happen in

n

k

ways.


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Exercise 2

Let x be a binomial random variable with n = 10 and p= 0.4.Find these values:a) P(x=4)
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Exercise 2

b) P(x 4)


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Exercise 2

c) P(x > 4)


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Exercise 2

d) P(x 4)
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Shape of Binomial Distribution

For p= 0.5, the shape of the distribution is symmetric. As pget closer to 0 or 1, the shape of the distribution becomesskewed.


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Mean and Variance

*Mean=np.*Variance=np(1 p) = npq.*Standard deviation=

npq
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Exercise 2

e) Mean and variance.


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Exercise 3

US Public Health Service
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Poisson distribution

Poisson distribution is appropriate for a random variable thatcounts the number of occurrences of an event of interest in agiven time interval.
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Assumption

Poisson distribution assumes that events occur independently.
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Example

Traffic accidents in a day.Speed limit violations in an hour.

Customers arriving at a bank in a day.
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Poisson Distribution From Binomial Distribution
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Poisson Distribution

If =average number of events in an specified time interval

then chance that k events will happen in that time is

P(X = k) =k e

k!,

where k 0, 1, 2, 3, . . ..
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Mean and Variance

*Mean=.

*Variance= .*Standard deviation=

.
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Exercise 4

Let X be a Poisson random variable with mean = 2.Calculate the following probabilities:a) P(X=0)
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Exercise 4

b) P(X = 1)

c) P(X > 1)
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Exercise 4

d) P(X = 5)
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Exercise 5

Airport Safety: The increased number of small commuterplanes in major airports has heightened concern over air safety.An eastern airport has recorded a monthly average of five near

misses on landings and takeoffs in the past 5 years.1 Find the probability that during a given month there are

no near-misses on landings and takeoffs at the airport.

2 Find the probability that during a given month there are

five near-misses.3 Find the probability that there are at least five

near-misses during a particular month.
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Exercise 6

Twenty people are asked to select a number from 0 to 9.Eight of them choose a 4,5, or 6.

1 If the choice of any one number is as likely as any other,what is the probability of observing eight or more choicesof the numbers 4,5, or 6?

2 What conclusions would you draw from the results of part

a?
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Exercise 7

In a food processing and packaging plant, there are, on an

average, two packaging machine breakdowns per week.1 What is the probability that there are no machine

breakdowns in a given week?

2 Calculate the probability that there are no more than two

machine breakdowns in two weeks?
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Extra Problem

When arrival probabilities are modeled by a PoissonDistribution, the probability of 2 arrivals in 2 hours is not

equal to twice the probability of 1 arrival in 1 hour, why?
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Poisson Process vs. Poisson Distribution
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a4 discrete probability models

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