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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 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 In STATA di Binomial(3,2,0.7) di Binomial(n,k,p)

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(3,2,0.7) di Binomial(n,k,p) is the likelihood of winning 1 or less (See help binomial() and more generally help scalar and the click on define) So we were looking for 1- binomial(3,2,0.7)

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, but 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 GRETL: Tools/p-value finder/binomial

19. B.2 Probability Distributions Figure B.2 PDF of a continuous random variable If we have a continuous variable instead 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 Principles of Econometrics, 3rd Edition

26. 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

27. 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

28. B.3.3 A Simple Experiment 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