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Chapter 2 – DIscrete DIstrIbutIons hüseyin güler

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Chapter 2 – DIscreteDIstrIbutIonshüseyingüler

MATHEMATICAL STATISTICS

DiscreteDistributions

- 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

- If A is a subset of S then

- Computetheprobabilitythatthenumber on thechip is 3 or 4.

Discrete Distributions

- Computetheprobabilitythatthenumber on thechip is lessthanorequalto 3.

Discrete Distributions

- 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

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

Discrete Distributions

- The weighted average of X is

- calledthemean of X.

- It is possibletoestimateμusingrelativefrequencies.

Discrete Distributions

- x1, x2,..., xn: Observedvalues of x
- fj: Thefrequency of uj
- uj = 1, 2, 3, 4.

- theempiricaldistribution
themean of theempiricaldistributionorthesamplemean

Discrete Distributions

- The variance of X is

- Thestandart deviationof X is

Discrete Distributions

- r_th moment abouttheorigin

Discrete Distributions

Discrete Distributions

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

Discrete Distributions