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### Appendix B

Econometrics

Memorial University of Newfoundland

Review of Probability Concepts

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 DistributionsFor 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-14

And 0.343 is the likelihood of winning exactly 3 games

B.2 Probability DistributionsFor 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-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)

B.2 Probability DistributionsFor 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-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)

B.2 Probability DistributionsFor 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-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

B.2 Probability DistributionsFor 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-18

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

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