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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

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

- B.1 Random Variables

Principles of Econometrics, 3rd Edition

- 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

- 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

Principles of Econometrics, 3rd Edition

Figure B.1 College Employment Probabilities

Principles of Econometrics, 3rd Edition

- 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

Principles of Econometrics, 3rd Edition

- 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

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

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

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:

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

Principles of Econometrics, 3rd Edition

Slide B-14

And 0.343 is the likelihood of winning exactly 3 games

Principles of Econometrics, 3rd Edition

Slide B-15

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)

Principles of Econometrics, 3rd Edition

Slide B-16

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)

Principles of Econometrics, 3rd Edition

Slide B-17

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

Principles of Econometrics, 3rd Edition

Slide B-18

Try instead:

http://www.zweigmedia.com/ThirdEdSite/stats/bernoulli.html

GRETL: Tools/p-value finder/binomial

Figure B.2 PDF of a continuous random variable

If we have a continuous

variable instead

Principles of Econometrics, 3rd Edition

Principles of Econometrics, 3rd Edition

Principles of Econometrics, 3rd Edition

Principles of Econometrics, 3rd Edition

Principles of Econometrics, 3rd Edition

Principles of Econometrics, 3rd Edition

Principles of Econometrics, 3rd Edition

- 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

Y = 1 if shaded Y = 0 if clear

X = numerical value (1, 2, 3, or 4)

Principles of Econometrics, 3rd Edition

Principles of Econometrics, 3rd Edition

Principles of Econometrics, 3rd Edition

Principles of Econometrics, 3rd Edition