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Basic Probability & Random Variables

Dive into the foundations of probability theory, including essential concepts such as mutually disjoint events, conditional probability, and the multiplicative rule. Discover the significance of Bayes' Rule in making informed inferences, particularly in contexts like estimating hidden behaviors. Explore random variables, their distributions, and how probability manifests in real-world scenarios, such as sampling methods in research. This comprehensive overview will equip you with the essential tools and knowledge to navigate the world of probability and statistical analysis effectively.

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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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