# Chapter 2 – DIscrete DIstrIbutIons hüseyin güler - PowerPoint PPT Presentation

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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 – DIscrete DIstrIbutIons hüseyin güler

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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

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

Discrete Distributions

### ThevarIance of theempIrIcaldIstrIbutIon

Discrete Distributions

### ThevarIanceandthe standart devIatIon of thesample

• s2 (the variance of the sample) is an estimate of (the variance of X).

Discrete Distributions