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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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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

  • 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 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 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 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 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 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 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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