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 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

slide3

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

slide4

P(X = x): RepresentstheprobabilitythatX is equaltox.

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

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

slide9

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 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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