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## Appendix B

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**ECON 4550**Econometrics Memorial University of Newfoundland Review of Probability Concepts Appendix B Adapted from Vera Tabakova’s notes**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**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**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**B.2 Probability Distributions**Principles of Econometrics, 3rd Edition**B.2 Probability Distributions**Figure B.1 College Employment Probabilities Principles of Econometrics, 3rd Edition**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**B.2 Probability Distributions**Principles of Econometrics, 3rd Edition**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**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**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**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:**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**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**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)**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)**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**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**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**B.2 Probability Distributions**Principles of Econometrics, 3rd Edition**B.3 Joint, Marginal and Conditional Probability**Distributions Principles of Econometrics, 3rd Edition**B.3 Joint, Marginal and Conditional Probability**Distributions Principles of Econometrics, 3rd Edition**B.3.1 Marginal Distributions**Principles of Econometrics, 3rd Edition**B.3.1 Marginal Distributions**Principles of Econometrics, 3rd Edition**B.3.2 Conditional Probability**Principles of Econometrics, 3rd Edition**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**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**B.3.3 A Simple Experiment**Principles of Econometrics, 3rd Edition**B.3.3 A Simple Experiment**Principles of Econometrics, 3rd Edition**B.3.3 A Simple Experiment**Principles of Econometrics, 3rd Edition