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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

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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
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
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
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 distributions5
B.2 Probability Distributions

Principles of Econometrics, 3rd Edition

b 2 probability distributions6
B.2 Probability Distributions

Figure B.1 College Employment Probabilities

Principles of Econometrics, 3rd Edition

b 2 probability distributions7
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 distributions8
B.2 Probability Distributions

Principles of Econometrics, 3rd Edition

b 2 probability distributions9
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 distributions10
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 distributions11
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 distributions12
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 distributions13
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 distributions14
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 distributions15
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 distributions16
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 distributions17
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 distributions18
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 distributions19
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 distributions20
B.2 Probability Distributions

Principles of Econometrics, 3rd Edition

b 3 joint marginal and conditional probability distributions
B.3 Joint, Marginal and Conditional Probability Distributions

Principles of Econometrics, 3rd Edition

b 3 joint marginal and conditional probability distributions22
B.3 Joint, Marginal and Conditional Probability Distributions

Principles of Econometrics, 3rd Edition

b 3 1 marginal distributions
B.3.1 Marginal Distributions

Principles of Econometrics, 3rd Edition

b 3 1 marginal distributions24
B.3.1 Marginal Distributions

Principles of Econometrics, 3rd Edition

b 3 2 conditional probability
B.3.2 Conditional Probability

Principles of Econometrics, 3rd Edition

b 3 2 conditional probability26
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
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 experiment28
B.3.3 A Simple Experiment

Principles of Econometrics, 3rd Edition

b 3 3 a simple experiment29
B.3.3 A Simple Experiment

Principles of Econometrics, 3rd Edition

b 3 3 a simple experiment30
B.3.3 A Simple Experiment

Principles of Econometrics, 3rd Edition

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