Chapter 2 â DIscrete DIstrIbutIons hÃ¼seyin gÃ¼ler

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# Chapter 2 â DIscrete DIstrIbutIons hÃ¼seyin gÃ¼ler - PowerPoint PPT Presentation

Chapter 2 – DIscrete DIstrIbutIons hüseyin güler. MATHEMATICAL STATISTICS. 2 . 1 . DIscrete ProbabIlIty DIstrIbutIons. The concept of random variable : S : Space or support of an experiment A random variable (r.v.) X is a real valued function defined on the space . X : S → R

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### Chapter 2 – DIscreteDIstrIbutIonshüseyingüler

MATHEMATICAL STATISTICS

DiscreteDistributions

2.1. DIscreteProbabIlItyDIstrIbutIons
• Theconcept of randomvariable:
• S: Spaceorsupport of an experiment
• A randomvariable (r.v.)X is a realvaluedfunctiondefined on thespace.
• X: S → R
• x: Representsthevalue of X
• xεS
• X is a discrete r.v. ifitspossiblevaluesarefinite, orcountablyinfinite.

Discrete Distributions

A chip is selectedrandomlyfromthebowl:

• S = {1, 2, 3, 4}
• X: Thenumber on theselectedchip
• X is a r.v. withspaceS
• x = 1, 2, 3, 4. X is a discrete r.v. (it takes 4 differentvalues)

Discrete Distributions

P(X = x): RepresentstheprobabilitythatX is equaltox.

• Thedistribution of probability on thesupportS
• Theprobabilitymassfunction (p.m.f.)

Discrete Distributions

CalculatIngprobabIlItIesusIngp.m.f.
• If A is a subset of S then
• Computetheprobabilitythatthenumber on thechip is 3 or 4.

Discrete Distributions

CalculatIngprobabIlItIesusIngp.m.f.
• Computetheprobabilitythatthenumber on thechip is lessthanorequalto 3.

Discrete Distributions

RelatIveFrequencIesandRelatIveFrequencyHIstogram
• When the experiment is performed n times the relative frequency of x is
• Thehistogram of relativefrequencies iscalledrelativefrequencyhistogram.
• Relativefrequenciesconvergetothe p.m.f as nincreases.

Discrete Distributions

Thechipexperiment is repeatedn = 1000 timesusing a computersimulation.

Discrete Distributions

ThecomparIson of f(x)andh(x)
• f(x) is theoreticallyobtainedwhileh(x) is obtainedfrom a sample.

Discrete Distributions

Themean of the (probabIlIty) dIstrIbutIon
• The weighted average of X is
• calledthemean of X.
• It is possibletoestimateμusingrelativefrequencies.

Discrete Distributions

Themean of theempIrIcaldIstrIbutIon
• x1, x2,..., xn: Observedvalues of x
• fj: Thefrequency of uj
• uj = 1, 2, 3, 4.
• theempiricaldistribution

themean of theempiricaldistributionorthesamplemean

Discrete Distributions

ThevarIanceandthestandarddevIatIon of thedIstrIbutIon
• The variance of X is
• Thestandart deviationof X is

Discrete Distributions

AN AlternatIveforthevarIance of thedIstrIbutIon