04 Discrete Probability Distribution MMC_.pdf

7/27/2019 04 Discrete Probability Distribution MMC_.pdf

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Math801MathematicalMethodsforComputing

DiscreteProbabilit Distribution

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ConceptofRandomVariable

DiscreteProbabilityDistribution

TheBinomial

Distribution

Home

Work

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Conceptof

RandomVariable

Inmostprobabilityproblems,weare

interestedin

one

number

that

is

associated

withtheoutcomeofexperiment.Such

,

outcomeof

experiment,

is

called

Random

.

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Examplesof

RandomVariable

Problems

1. Inmanufacturing,x=No.ofdefectiveitems

= .

,

3. Throwingtwodice,z =sumoftwodice

situationsinvolvingrandomness,therefore,

a o ea ovevar a ex,y,an zare reedifferentrandomvariables.

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StudyingRandomVariablesweinterestin:

valuesintheirrange.T is

sprea

o

pro a i ities

or

t e

various

valuesofrandomvariablesiscalleda

probabilitydistribution.Inthefollowing,weconstructanexample:

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Experimentof

Throwingtwo

Dice

Results:

Tabulated:

Thinkabouthavinga

6

sumo wo ce=z=

P(X=7)=6/36


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Pro a i ity istri ution unction p ort esum

oftwo

dice

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Note:Abovediagramisexperimental,makeyourselfidealpdf.


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distribution

calledprobability

densityfunction

orprobability

massfunction.

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Whentherandomvariabletakesthediscrete

values,its

robabilit

distribution

will

be

DiscreteProbabilityDistribution.

,

defectiveitemsineachshipment willhave

discreteprobabilitydistributions.

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Byunderstandingtheunderlyingpattern

behindeach

ex eriment,

we

have

formulated

differentprobabilitydistribution.

,

importantdiscreteprobabilitydistribution

calledTheBinomialProbabilityDistribution.

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Considerexperimentoftossingacoin:

.

2. Successprobability,p,remainssame

3. Thesuccessive

trials

are

all

independent

threeconditionsarecalledBernoullitrails.Theexperimenthavingn Bernoullitrialsis

calledabinomialprobabilityexperiment.11


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WhenXdenotesthenumberofsuccessesinn

trialsof

abinomial

robabilit

ex eriment,

it

iscalledabinomialrandomvariablehaving

. . .

X, thebinomial

p.d.

is

given

by

the

formula:

( ; , ) ( ) , 0,1,2,...,x n x

n

b x n p P X x p q x n

-

= = = =

where q =1-p is

the

probability

of

failure 12


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Q:Acoinistossed5times.Whatareprobabilities

ofobtainin

various

number

of

heads?

SincetheseareBernoullitrialswithn=5times.

. .

withp=1/2,q=1/2,andn=5.Thereforeweapply

x n xn - , , , , ,...,x

= = = =

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x n xn

-

MainformulaforBPD

, , , , ,...,x

551 1 1

( ; 5 , ) ( ) , 0,1,2, 3, 4, 5

x x

b x P X x x x

- = = = =

0 5 051 1 1 1 1- ; ,

02 2 2 32 32= = = = =

51 1 1 1 1 10(3; 5 , ) ( 3) 10

32 2 2 8 4 32b P X

- = = = = =

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.

MakeaMatlab programtoimplement

( ; , ) ( ) , 0,1,2,...,x n xn

b x n p P X x p q x n x

- = = = =

suchthat,

for

any

input

of

nand

p,

program

enerateahisto ram.

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Questions?

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04 Discrete Probability Distribution MMC_.pdf