Review of Probability Concepts 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 X is 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 variable X is the number of successes in n independent trials of identical experiments with probability of success p. (B.1) Principles of Econometrics, 3rd Edition

B.2 Probability Distributions 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 Principles of Econometrics, 3rd Edition Slide B-10 10

B.2 Probability Distributions 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 Principles of Econometrics, 3rd Edition Slide B-11 11

B.2 Probability Distributions 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: times Principles of Econometrics, 3rd Edition Slide B-12 12

B.2 Probability Distributions 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 Principles of Econometrics, 3rd Edition Slide B-13 13

B.2 Probability Distributions 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 Principles of Econometrics, 3rd Edition Slide B-14 14

B.2 Probability Distributions 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 Principles of Econometrics, 3rd Edition Slide B-15 15

B.2 Probability Distributions 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) Principles of Econometrics, 3rd Edition Slide B-16 16

B.2 Probability Distributions 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 Principles of Econometrics, 3rd Edition Slide B-17 17

B.2 Probability Distributions 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 Principles of Econometrics, 3rd Edition Slide B-18 18

B.2 Probability Distributions Figure B.2 PDF of a continuous random variable A continuous variable 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 Check error In posted slides!!! Principles of Econometrics, 3rd Edition

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

B.3.2 Conditional Probability What is the likelihood Of having an income If the person has a 4-year degree? y .04/.18=.22 1 .14/.18=.78 Principles of Econometrics, 3rd Edition

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 y .04/.18=.22 1 .14/.18=.78 Principles of Econometrics, 3rd Edition Slide B-26 26

B.3.2 Conditional Probability Two random variables are statistically independent if the conditional probability that Y = y given that X = x, is the same as the unconditional probability that Y = y. (B.4) (B.5) (B.6) 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