Engineering StatisticsDiscrete Probability Distributions

Course outline

� Random Variables

� Discrete Probability Distributions

� Expected Value and Variance

� Binomial Probability Distribution

� Poisson Probability Distribution

� Hypergeometric Probability Distribution

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0,05

0,1

0,15

0,2

0,25

0,3

0,35

1 2 3 4 5 6

Random Variable

� A random variable is a numerical description of the outcome

of an experiment.

� The particular numerical value of the random variable

depends on the outcome of the experiment.

� A random variable can be classified as being either discrete

or continuous depending on the numerical values it

assumes.

� A discrete random variable may assume either a finite

number of values or an infinite sequence of values, such as

0,1,2,3……

� A continuous random variable may assume any numerical

value in an interval or collection of intervals, for example

0<X<203

Examples of discrete random variables

Experiment Random Variable (X)

Possible Values

for the Random

Variable

Contact five

customers

Inspect

shipment of

50 pumps

Operate a toll

road in a day

Ferry services

in a day

Number of customers

who place an order

Number of defective

pumps

Number of cars arriving

the tollbooth in a day

Number of passengers

using the ferry services in

a day

0,1,2,3,4,5

1,2,3,…,49,50

1,2,3,…..

1,2,3,…..

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Discrete random variable

with a finite number of

values

Discrete random variable

with an infinite number

Sequence of values

Continuous Random Variables

� A random variable is continuous if it can assume

any numerical value in an interval or collection of

intervals

� Experimental outcomes that can be described by

continuous random variables:

� Heights, weights, temperature, etc

� Waiting times

� Measurement errors

� Length of life of a particular equipment

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Discrete Probability Distributions

� The probability distribution for a random variable describes

how probabilities are distributed over the values of the

random variable.

� The probability distribution is defined by a probability

function, denoted by f(x), which provides the probability for each value of the random variable.

� The required conditions for f(x):

f(x) > 0

Σf(x) = 1

� We can describe a discrete probability distribution with a

table, graph, or equation.

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Example

� Probability distribution for the number of

automobiles sold during a day at DicarloMotors

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Example

� Graphical Representation of the Probability

Distribution

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Discrete Probability Distribution

� Discrete uniform probability distribution

� Binomial probability distribution

� Poisson probability distribution

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Discrete Uniform Probability Distribution

� The discrete uniform probability distribution is the simplest

example of a discrete probability distribution given by a

formula.

� The discrete uniform probability function is:

� Note that ALL values of the random variable are equally likely.

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Expected Value and Variance

� The expected value, or mean, of a random variable is a measure of its central location.

� The variance summarizes the variability in the values of a random variable.

� The standard deviation, σ, which is the positive square root of the variance.

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Example

� Expected Value of a Discrete Random Variable

Calculation of the expected value for the number of

automobiles sold during a day at Dicarlo Motors

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Example

� Variance and Standard Deviation of a Discrete

Random Variable

Calculation of the variance for the number of automobiles

sold during a day at DicarloMotors

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Binomial Probability Distribution

� Binomial Distribution is one of the most important discrete

distributions. It is associated with a multiple-step experiment

called Binomial Experiment.

� Properties of a Binomial Experiment

1. The experiment consists of a sequence of n identical

trials.

2. Two outcomes, success and failure, are possible on

each trial.

3. The probability of a success, denoted by p, does not

change from trial to trial. (p is fixed)

4. The trials are independent.

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Binomial Probability Function

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Example: Martin Clothing Store Problem

Let us consider the purchase decisions of the next

three customers who enter the Martin Clothing

Store. On the basis of past experience, the store

manager estimates the probability that any one

customer will make a purchase is 0,30.

What is the probability that two of the next three

customers will make a purchase?

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Example: Martin Clothing Store Problem

Using Tree� S to denote success (a purchase) and F to denote failure (no

purchase), we are interested in experimental outcomes involving

two successes in the three trials (purchase decisions).

� Checking the four requirements for a binomial experiment, we

note that:

� The experiment can be described as a sequence of three identical trials, one

trial for each of the three customers who will enter the store.

� Two outcomes – the customer makes a purchase (success) or the customer

does not make a purchase (failure) – are possible for each trial.

� The probability that the customer will make a purchase (0,30) or will not

make a purchase (0,70) is assumed to be the same for all customers.

� The purchase decision of each customer is independent of the decisions of

the other customers

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Example: Martin Clothing Store Problem

Using Tree Diagram

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Example: Martin Clothing Store Problem

Using Tree Diagram

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Example: Martin Clothing Store Problem

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Example: Martin Clothing Store Problem

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Example: Martin Clothing Store Problem

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Example: Martin Clothing Store Problem

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Example: Martin Clothing Store Problem

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Example

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Binomial Probability Distribution

� Expected Value

E(x) = µ = np

� Variance

Var(x) = σ 2 = np(1 - p)

� Standard Deviation

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)1()(SD pnpx −== σ

Example: Martin Clothing Store problem

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Example

� Sebuah pabrik menemukan bahwa secara rata-

rata 20% dari baut yang diproduksi oleh sebuah

mesin akan mengalami penyimpangan dari

persyaratan yang dispesifikasikan (cacat). Jika 10

baut dipilih secara acak dari produksi harian

mesin ini, maka hitunglah probabilitas:

� Tepat dua baut akan cacat

� Minimal dua baut akan cacat

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Poisson Probability Distribution

� The Poisson random variable X is used for

determining the number of occurrences of

specified event in a particular time interval or

space.

� Examples:

� Number of arrivals at a car wash in one hour

� Number of machine breakdowns in a day

� Number of repairs needed in 10 miles of highway

� Number of leaks in 100 miles of pipeline.

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Poisson Probability Distribution

� Properties of a Poisson Experiment

1. The probability of an occurrence is the same for any two

intervals of equal length.

2. The occurrence or non-occurrence in any interval is

independent of the occurrence or non-occurrence in any

other interval.

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Example

� Using the Poisson Probability Function

The average machine breakdowns during their operation is

three per week. Find the probability of exactly one machine

breakdown during a week.

Other time interval also can be used. For example, what is

exactly one machine breakdown during two weeks?

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���� =� � � −�

�!=

31� −3

1!

= 3�−3 = 0,1493

Example

� Using the Table of Poisson Probabilities

The average machine breakdowns during their operation is three

per week. Find the probability of exactly one machine breakdown

during a week.

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Example

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Unit consistency in probabilities calculation

� Example:

� Average machine breakdown during their

operation is three per week

� Average machine breakdown during their

operation within two weeks is six machines

� Average machine breakdown during their

operation within five weeks is fifteen machines

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Example

� Using the Poisson Probability Function

The average machine breakdowns during their operation is

three per week. Find the probability of exactly 10 machines

breakdown during five weeks.

� � =�����

�!=

15�����

10!= 0.0486

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Example

� Karena kurangnya pengawasan saat penggelaran pipa,

diketahui bahwa kejadian kebocoran pipeline diketahui

mengikuti distribusi Poisson dengan rata-rata adalah 0,7

kebocoran setiap 10 km pipa. Dari informasi tersebut

hitunglah:

� Peluang dalam 20 km pipa terjadi kurang dari 2 kebocoran.

� Peluang dalam 20 km pipa terjadi lebih dari 2 kebocoran.

� Peluang dalam 50 km pipa tidak terjadi kebocoran pipa sama

sekali.

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Hypergeometric Probability Distribution

� The hypergeometric distribution is closely related to the

binomial distribution.

� With the hypergeometric distribution, the trials are not

independent, and the probability of success changes from

trial to trial.

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Mean and Variance

� Mean

� Variance

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Example: Ontario Electric

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Example: Ontario Electric

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Example: Ontario Electric

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References

� Anderson, Sweeney, Williams., Statistics for

Business and Economics, 11th Edition., West

Publishing Company, 2011.

� Statistics for Business and Economics., Slides

Prepared by John S. Loucks St. Edward’s

University,South-Western/Thompson Learning

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