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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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Appendix B

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Review of probability concepts l.jpg

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 l.jpg

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 l.jpg

    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 l.jpg

    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


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    B.2 Probability Distributions

    Principles of Econometrics, 3rd Edition


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    B.2 Probability Distributions

    Figure B.1 College Employment Probabilities

    Principles of Econometrics, 3rd Edition


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    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 l.jpg

    B.2 Probability Distributions

    Principles of Econometrics, 3rd Edition


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    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 l.jpg

    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 l.jpg

    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 l.jpg

    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 l.jpg

    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 l.jpg

    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 l.jpg

    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 l.jpg

    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 l.jpg

    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


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    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


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    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 l.jpg

    B.2 Probability Distributions

    Principles of Econometrics, 3rd Edition


    B 3 joint marginal and conditional probability distributions l.jpg

    B.3 Joint, Marginal and Conditional Probability Distributions

    Principles of Econometrics, 3rd Edition


    B 3 joint marginal and conditional probability distributions22 l.jpg

    B.3 Joint, Marginal and Conditional Probability Distributions

    Principles of Econometrics, 3rd Edition


    B 3 1 marginal distributions l.jpg

    B.3.1 Marginal Distributions

    Principles of Econometrics, 3rd Edition


    B 3 1 marginal distributions24 l.jpg

    B.3.1 Marginal Distributions

    Principles of Econometrics, 3rd Edition


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    B.3.2 Conditional Probability

    Principles of Econometrics, 3rd Edition


    B 3 2 conditional probability26 l.jpg

    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 l.jpg

    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 l.jpg

    B.3.3 A Simple Experiment

    Principles of Econometrics, 3rd Edition


    B 3 3 a simple experiment29 l.jpg

    B.3.3 A Simple Experiment

    Principles of Econometrics, 3rd Edition


    B 3 3 a simple experiment30 l.jpg

    B.3.3 A Simple Experiment

    Principles of Econometrics, 3rd Edition


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