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Basic Probability & Random Variables

Basic Probability & Random Variables. Axioms of Probability. If are mutually disjoint , then. Conditional Probability. Multiplicative Rule of Probability. BAYES RULE:. Bayes Rule is very important. Why? Often we want to know But what we do know is

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Basic Probability & Random Variables

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  1. Basic Probability & RandomVariables

  2. Axioms of Probability • Ifaremutuallydisjoint, then

  3. ConditionalProbability

  4. MultiplicativeRule of Probability BAYES RULE:

  5. BayesRule is veryimportant Why? • Oftenwewanttoknow • But whatwe do know is • Wewill be abletoinfer by

  6. Here is a usefulapplication of Bayes From thegraphwe can seethat,

  7. RandomizationResponseTheory Assumethatyouneedestimatetheproportion of narcoticdrugconsumptionamonguniversitystudents. It is unlikelythatstudentswouldansweryourquestionnairehonestly. So here is a simpletrickyoumayuse. Instead of askingthequestiondirectly, letthestudentdraw a ballfrom an urn in whichthereare 8 blueand 2 yellowballs. If a yellowball is drawn (you do not seetheresult), studentanswersthequestion «Is thelastdigit of your TC ID numberodd?» andif a blueball is drawnthenthestudentanswers «Haveyou ever used a narcoticdrug?» question.

  8. Assume youhaveaskedthisquestionto 17 studentsand 13 of themanswered YES. Soonwewill be abletocomputetheerrordueto …(?)

  9. RandomVariablesandProbabilityDistributions • RandomVariable:? Example: Whenrolling a twodice, wemay be interested in whetheror not thesum of thetwodice is 7. Orwemight be interested in thesum of thetwodice.

  10. Example: How longdoes it takeforthenextbustoarrive?

  11. Now, suppose the probability that the T comes in any given minute is a constant , and whether the T comes is independent of what has happened in previous periods. • What's P(X=1)? • What's P(X=2)? • What’s P(X=3)? • What’s P(X=x)? Geometric Distribution with a parameter

  12. ProbabilityDensityFunction • An alternative model where Y is exact time: If Inclass: How probabilitiesarerelatedwithareasunderthecurve.

  13. Expectation • Discrete Case • Continuous Case

  14. Discrete • Continuous

  15. Variance

  16. Typicallyyouneedtoknowwhatsort of probabilitydistributionsarethereandforwhichtype of situationsthayareusedfor. • Wewill be mostlydealingwith Normal Distribution.

  17. INCOME DISTRIBUTION – (Empirical)

  18. INCOME DISTRIBUTION – (Theoretical) Log Normal Didtribution

  19. s x m Normal Distribution • Normal distribution has an unfriendly form that does not let explicit integration: • However any normal distribution can be transformed into standard normal distribution Standard Normal Distribution Normal Distribution s=1 z m=0

  20. Normal Distribution Standard Normal Distribution μ = 500 σ = 100 μ = 0 σ = 1 P(x < 600) P(z < 1) z x Same Area μ =500 600 μ = 0 1 P(x < 500) = P(z < 1)

  21. Before going any further did you notice that statistical parameters are actually operational definitions for some concepts. • Let’s discuss these operationalized variables and their corresponding concepts:

  22. Sampling Probability Sampling Nonprobability Sampling

  23. Probability Sampling • Sampling element • Population • Target population • Sampling frame • Sampling ratio

  24. There is a classic Jimmy Stewart movie, Magic Town, about "Grandview," a small town in the Midwest that is a perfect statistical microcosm of the United States, a place where the citizens' opinions match perfectly with Gallup polls of the entire nation. A pollster (Jimmy Stewart), secretly uses surveys from this "mathematical miracle" as a shortcut to predicting public opinion. Instead of collecting a national sample, he can more quickly and cheaply collect surveys from this single small town. The character played by Jane Wyman, a newspaper editor, finds out what is going on and publishes her discovery. As a result the national media descend upon the town, which becomes, overnight, "the public opinion capital of the U.S."

  25. Probability Sampling

  26. Check http://www.socialresearchmethods.net POPULATION PARAMETERS SAMPLE STATISTICS To be filled in class

  27. Sampling Distribution

  28. Probability Sampling • Random sample • Sampling error • Four Ways to Sample Randomly • Simple Random • Systematic • Stratified Sampling • Cluster Sampling

  29. Random Sample Variation Component • Sampling Error: Sample size Component

  30. Sampling Distribution and Sampling Error Let’s first see what mathematics have to say. According to Law of Large Numbers: As sample size increases (approaches to ) sample mean approaches to population mean, in mathematical symbols According to Central Limit Theorem As the number of samples (not the sample size, this time) increases then sample mean has a normal distribution with mean andstandarddeviation. Mathematically we say,

  31. Sampling and Confidence x Confidence information is in z. can be replacedby.

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