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

2.  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

3.  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.

4.  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.

6. Figure B.1 College Employment Probabilities

7.  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

9.  For example, a binomial random variable X is the number of successes in n independent trials of identical experiments with probability of success p.

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.  First: for winning only once in three weeks, likelihood is 0.189, see?  Times

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.  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.  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.  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.  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.  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.  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/be rnoulli.html http://www.zweigmedia.com/ThirdEdSite/stats/be rnoulli.html  GRETL: Tools/p-value finder/binomial

19. Figure B.2 PDF of a continuous random variable If we have a continuous variable instead

25. y 0.04/.18=.22 1.14/.18=.78

26.  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.

27. Y = 1 if shaded Y = 0 if clear X = numerical value (1, 2, 3, or 4)