1 / 16

Chapter 2 – DIscrete DIstrIbutIons hüseyin güler

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

paiva
Download Presentation

Chapter 2 – DIscrete DIstrIbutIons hüseyin güler

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 2 – DIscreteDIstrIbutIonshüseyingüler MATHEMATICAL STATISTICS DiscreteDistributions

  2. 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

  3. 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

  4. P(X = x): RepresentstheprobabilitythatX is equaltox. • Thedistribution of probability on thesupportS • Theprobabilitymassfunction (p.m.f.) Discrete Distributions

  5. Discrete Distributions

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

  7. CalculatIngprobabIlItIesusIngp.m.f. • Computetheprobabilitythatthenumber on thechip is lessthanorequalto 3. Discrete Distributions

  8. 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

  9. Thechipexperiment is repeatedn = 1000 timesusing a computersimulation. Discrete Distributions

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

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

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

  13. ThevarIanceandthestandarddevIatIon of thedIstrIbutIon • The variance of X is • Thestandart deviationof X is Discrete Distributions

  14. AN AlternatIveforthevarIance of thedIstrIbutIon • r_th moment abouttheorigin Discrete Distributions

  15. ThevarIance of theempIrIcaldIstrIbutIon Discrete Distributions

  16. ThevarIanceandthe standart devIatIon of thesample • s2 (the variance of the sample) is an estimate of (the variance of X). Discrete Distributions

More Related