The Exponential, Gamma and Chi-Squared

8/13/2019 The Exponential, Gamma and Chi-Squared

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1

Lecture 12:The Exponential, Gamma

and Chi-Squared PDFsDevore, Ch. 4.4


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Topics

I. Properties of Gamma Distribution

II.Computing Probabilities with Gamma

III.Exponential Distribution andApplications

IV.Chi-Squared Distribution


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I. The Gamma Distribution

Gamma familyrepresents a variety of skeweddistributions based on a shape and scale parameter.

Shape parameter~ a; Scale parameter ~ b

0.0

0.2

0.4

0.6

0.8

1.0

0

2

4

6

8

f(x;

)

a=2; b=1/3

a=1; b=1

a=2; b=1

a=2; b=2

Scale Parameter:b< 1 compresses f(x) in xb = 1 standard gammab> 1 stretches f(x) in x

Shape Parameter:a 1 f(x) strictlydecreasesa> 1 f(x) rises to amaximum and thendecreases


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5

The Gamma and the StandardGamma PDFs

Gamma distribution pdf Satisfies both conditions of a pdf.

If atakes an integer valuegamma becomes theErlang distribution

Gamma Distribution (b>0) Standard Gamma Distribution (b=1)

Note: a > 0; b > 0 E(X) = m = ab

V(X) = s2

= ab

2

( ) ( )

G=--

otherwise

xexxf

x

0

01

,;1 ba

a abba ( ) ( )

G=--

otherwise

xexxf

x

0

01

,;1a

aba


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II. Computing Probs Using Gamma

Suppose X is a continuous rv, then the cdf for thestandard gamma rv is:

Above equation also known as incomplete gammafunction. Cumulative Gamma Tables: A.4 (Page 742)

For non-standard Gamma, the cdf can be found as:

0 X

)(

);(1

0

>

G

=--

dyey

xFyx

a

aa

( ) functiongammaincompletetheis;where

;),;(

a

ab

ba

=

F

xFxF


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Gamma Applications

Reliability Assessment, Queuing Theory, ComputerEvaluations, Biological Studies

X represents the time of occurrence of an event thatdepends on a series of independent sub-events.

Example: suppose emails are sent to a centralprocessor and then are released in batches.

Let X equal the time (in seconds) that it takes for an emailmessage to be released after arrival at the centralprocessor. Lets assume X follows a gamma distribution:

Shape parameter = 4.

Scale parameter = 2.

What is the expected time between batches?

What is the prob that an individual message will wait morethan 20 sec before being sent from the central processor?

95% of messages will take longer than how many secs?


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Useful Excel / Minitab Commands

Excel

cdf =gammadist(x,a,b,true)

=GAMMADIST(20,4,2,TRUE)

Inverse =gammainv(prob,a,b)

=GAMMAINV(0.05,4,2)

Minitab

CALC >> Prob Distributions


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III. Exponential Distribution

Exponential pdf is a special case of gamma pdf

where the shape parameter, a= 1.

Derive Exponential from Gamma (a= 1)

If you letl= 1/b then X is exponential if pdf

f(x; l) = ?

( ) ( )

G=--

otherwise

xexxf

x

0

01

,;1 ba

a abba

( )

>

=-

otherwise

xexf

x

0

0,0;

lll

l


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Computer Processor Example

Solve the prior example (slide 7) assuming that e-mails are sent immediately upon arrival at centralprocessor assuming that the shape parameter equals

1 and the scale parameter equals 8.

Define X

What isl= ?

What is the probability that the time betweenmessages sent from the central processor willexceed 20 sec?

Compare this answer with the batch process?


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Constant Failure Rate Model

One application of the exponential distribution occurs in

reliability (relates to Poisson distribution).

Many products are assumed to have constant failurerates (average time between failures, l).

Under a constant failure rate model, there is no

wearout,so that the distribution of additional lifetimeat any time is exactly the same as when new

(memoryless property!!).


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Distribution of Failures

What distribution function models the # of failures per time period iffailure rate is assumed constant? So, we have the same rate per timeperiod (example: 0.1/year)

0.0

20.0

40.0

60.0

80.0

100.0

120.0

0 2 4 6 8 10 12 14 16 18 20

time period, t

#ofFailures

Poisson models # of

failures, whileexponential modelsthe time betweensuccessivefailures!


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Reliability Prediction -Exponential

Given a constant failure rate (0.1 failures/yr),

what is the probability that a unit will experience

its first failure after 10 years?

1);()Pr( tetFtT ll --==

R(t) = probability that the time to failure is greaterthan time t.

So, R(t) = 1 P(T


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R(t) - Reliability Function

Probability unit will not fail in first 6 months = .9512

Probability unit will not fail after 2 years = .8

What is R(10)?

Time

Period R(t)

0.5 0.9512

1 0.9048

1.5 0.8607

2 0.8187

2.5 0.7788

3 0.7408

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

time period, year

R(t)


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Service Calls Example

Suppose you work for service operation andreceive 0.5 calls per hour.

Let T equal the time between successive calls

Assume T ~ EXP(l=0.5)

What is the probability that more than 3 hourswill elapse between calls?

What is the probability that more than mean +3shours will elapse between calls?

What is the probability that less than the mean -3s hours will elapse between calls?


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The Exponential RV and itsmemoryless property

A very useful property of theexponential RV is its memorylessproperty.

Exponential and Poisson Number ofoccurrences ~ Poisson. Time betweenoccurrences ~ Exponential

Return to TV example:

Suppose your TV has lasted 2 years. What is

the probability it will last another year?

Hint: Find P(X >= t + to| X>= to)

M l P d


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Memoryless Property andUseful Life

Memoryless property is used to modelproducts treated as good as newuntil failure.

Can you think of any such products?

Does the constant failure rate modelbecome a less effective assumptionwith longer periods of time where

products start to degrade with age?


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The Exponential, Gamma and Chi-Squared