Chapter 2 discrete distributions h seyin g ler
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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 discrete distributions h seyin g ler

Chapter 2 – DIscreteDIstrIbutIonshüseyingüler

MATHEMATICAL STATISTICS

DiscreteDistributions


2 1 discrete probability distributions

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


Chapter 2 discrete distributions h seyin g ler

  • 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


Chapter 2 discrete distributions h seyin g ler

  • P(X = x): RepresentstheprobabilitythatX is equaltox.

  • Thedistribution of probability on thesupportS

  • Theprobabilitymassfunction (p.m.f.)

Discrete Distributions


Chapter 2 discrete distributions h seyin g ler

Discrete Distributions


Calculating probabilities using p m f

CalculatIngprobabIlItIesusIngp.m.f.

  • If A is a subset of S then

  • Computetheprobabilitythatthenumber on thechip is 3 or 4.

Discrete Distributions


Calculating probabilities using p m f1

CalculatIngprobabIlItIesusIngp.m.f.

  • Computetheprobabilitythatthenumber on thechip is lessthanorequalto 3.

Discrete Distributions


Relative frequencies and relative frequency histogram

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


Chapter 2 discrete distributions h seyin g ler

  • Thechipexperiment is repeatedn = 1000 timesusing a computersimulation.

Discrete Distributions


The comparison of f x and h x

ThecomparIson of f(x)andh(x)

  • f(x) is theoreticallyobtainedwhileh(x) is obtainedfrom a sample.

Discrete Distributions


The mean of the probability distribution

Themean of the (probabIlIty) dIstrIbutIon

  • The weighted average of X is

  • calledthemean of X.

  • It is possibletoestimateμusingrelativefrequencies.

Discrete Distributions


The mean of the empirical distribution

Themean of theempIrIcaldIstrIbutIon

  • x1, x2,..., xn: Observedvalues of x

  • fj: Thefrequency of uj

  • uj = 1, 2, 3, 4.

  • theempiricaldistribution

    themean of theempiricaldistributionorthesamplemean

Discrete Distributions


The variance and the standard deviation of the distribution

ThevarIanceandthestandarddevIatIon of thedIstrIbutIon

  • The variance of X is

  • Thestandart deviationof X is

Discrete Distributions


An alternative for the variance of the distribution

AN AlternatIveforthevarIance of thedIstrIbutIon

  • r_th moment abouttheorigin

Discrete Distributions


The variance of the empirical distribution

ThevarIance of theempIrIcaldIstrIbutIon

Discrete Distributions


The variance and the standart deviation of the sample

ThevarIanceandthe standart devIatIon of thesample

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

Discrete Distributions


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