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Review of Probability Concepts in Econometrics

This appendix provides a comprehensive review of probability concepts in econometrics, including random variables, probability distributions, joint and conditional probabilities, and properties of probability distributions. It also discusses some important probability distributions commonly used in econometric analysis.

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Review of Probability Concepts in Econometrics

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  1. ECON 4550 Econometrics Memorial University of Newfoundland Review of Probability Concepts Appendix B Adapted from Vera Tabakova’s notes

  2. Appendix B: Review of Probability Concepts • B.1 Random Variables • B.2 Probability Distributions • B.3 Joint, Marginal and Conditional Probability Distributions • B.4 Properties of Probability Distributions • B.5 Some Important Probability Distributions Principles of Econometrics, 3rd Edition

  3. B.1 Random Variables • A random variable is a variable whose value is unknown until it is observed. • A discrete random variable can take only a limited, or countable, number of values. • A continuous random variable can take any value on an interval. Principles of Econometrics, 3rd Edition

  4. B.2 Probability Distributions • The probability of an event is its “limiting relative frequency,” or the proportion of time it occurs in the long-run. • The probability density function (pdf) for a discrete random variable indicates the probability of each possible value occurring. Principles of Econometrics, 3rd Edition

  5. B.2 Probability Distributions Principles of Econometrics, 3rd Edition

  6. B.2 Probability Distributions Figure B.1 College Employment Probabilities Principles of Econometrics, 3rd Edition

  7. B.2 Probability Distributions • The cumulative distribution function (cdf) is an alternative way to represent probabilities. The cdf of the random variable X, denoted F(x),gives the probability that Xis less than or equal to a specific value x Principles of Econometrics, 3rd Edition

  8. B.2 Probability Distributions Principles of Econometrics, 3rd Edition

  9. B.2 Probability Distributions • For example, a binomial random variableX is the number of successes in n independent trials of identical experiments with probability of success p. Principles of Econometrics, 3rd Edition

  10. B.2 Probability Distributions Principles of Econometrics, 3rd Edition Slide B-10 For example, if we know that the MUN basketball team has a chance of winning of 70% (p=0.7) and we want to know how likely they are to win at least 2 games in the next 3 weeks

  11. B.2 Probability Distributions Principles of Econometrics, 3rd Edition Slide B-11 For example, if we know that the MUN basketball team has a chance of winning of 70% (p=0.7) and we want to know how likely they are to win at least 2 games in the next 3 weeks First: for winning only once in three weeks, likelihood is 0.189, see? Times

  12. B.2 Probability Distributions times Principles of Econometrics, 3rd Edition Slide B-12 For example, if we know that the MUN basketball team has a chance of winning of 70% (p=0.7) and we want to know how likely they are to win at least 2 games in the next 3 weeks… The likelihood of winning exactly 2 games, no more or less:

  13. B.2 Probability Distributions Principles of Econometrics, 3rd Edition Slide B-13 For example, if we know that the MUN basketball team has a chance of winning of 70% (p=0.7) and we want to know how likely they are to win at least 2 games in the next 3 weeks So 3 times 0.147 = 0.441 is the likelihood of winning exactly 2 games

  14. B.2 Probability Distributions Principles of Econometrics, 3rd Edition Slide B-14 For example, if we know that the MUN basketball team has a chance of winning of 70% (p=0.7) and we want to know how likely they are to win at least 2 games in the next 3 weeks And 0.343 is the likelihood of winning exactly 3 games

  15. B.2 Probability Distributions Principles of Econometrics, 3rd Edition Slide B-15 For example, if we know that the MUN basketball team has a chance of winning of 70% (p=0.7) and we want to know how likely they are to win at least 2 games in the next 3 weeks For winning only once in three weeks: likelihood is 0.189 0.441 is the likelihood of winning exactly 2 games 0.343 is the likelihood of winning exactly 3 games So 0.784 is how likely they are to win at least 2 games in the next 3 weeks

  16. B.2 Probability Distributions Principles of Econometrics, 3rd Edition Slide B-16 For example, if we know that the MUN basketball team has a chance of winning of 70% (p=0.7) and we want to know how likely they are to win at least 2 games in the next 3 weeks So 0.784 is how likely they are to win at least 2 games in the next 3 weeks In STATA di binomial(3,1,0.7) di Binomial(n,k,p) is the likelihood of winning 1 or less So we were looking for 1- binomial(3,1,0.7)

  17. B.2 Probability Distributions Principles of Econometrics, 3rd Edition Slide B-17 For example, if we know that the MUN basketball team has a chance of winning of 70% (p=0.7) and we want to know how likely they are to win at least 2 games in the next 3 weeks So 0.784 is how likely they are to win at least 2 games in the next 3 weeks In SHAZAM, although there are similar commands, it is a bit more cumbersome See for example: http://shazam.econ.ubc.ca/intro/stat3.htm

  18. B.2 Probability Distributions Principles of Econometrics, 3rd Edition Slide B-18 For example, if we know that the MUN basketball team has a chance of winning of 70% (p=0.7) and we want to know how likely they are to win at least 2 games in the next 3 weeks Try instead: http://www.zweigmedia.com/ThirdEdSite/stats/bernoulli.html

  19. B.2 Probability Distributions Figure B.2 PDF of a continuous random variable A continuous variable Principles of Econometrics, 3rd Edition

  20. B.2 Probability Distributions Principles of Econometrics, 3rd Edition

  21. B.3 Joint, Marginal and Conditional Probability Distributions Principles of Econometrics, 3rd Edition

  22. B.3 Joint, Marginal and Conditional Probability Distributions Principles of Econometrics, 3rd Edition

  23. B.3.1 Marginal Distributions Principles of Econometrics, 3rd Edition

  24. B.3.1 Marginal Distributions Principles of Econometrics, 3rd Edition

  25. B.3.2 Conditional Probability What is the likelihood Of having an income If the person has a 4-year degree? Principles of Econometrics, 3rd Edition

  26. B.3.2 Conditional Probability What is the likelihood Of having an income If the person has a 4-year degree? 78% But only 69% in general Principles of Econometrics, 3rd Edition Slide B-26

  27. B.3.2 Conditional Probability • Two random variables are statisticallyindependent if the conditional probability that Y = y given that X = x, is the same as the unconditional probability that Y = y. Principles of Econometrics, 3rd Edition

  28. B.3.3 A Simple Experiment Y = 1 if shaded Y = 0 if clear X = numerical value (1, 2, 3, or 4) Principles of Econometrics, 3rd Edition

  29. B.3.3 A Simple Experiment Principles of Econometrics, 3rd Edition

  30. B.3.3 A Simple Experiment Principles of Econometrics, 3rd Edition

  31. B.3.3 A Simple Experiment Principles of Econometrics, 3rd Edition

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