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Appendix B

ECON 6002 Econometrics Memorial University of Newfoundland. Review of Probability Concepts. Appendix B. SECOND. Adapted from Vera Tabakova’s notes. B.4 Properties of Probability Distributions. B.4.1 Mean, median and mode For a discrete random variable the expected value is:.

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Appendix B

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  1. ECON 6002 Econometrics Memorial University of Newfoundland Review of Probability Concepts Appendix B SECOND Adapted from Vera Tabakova’s notes

  2. B.4 Properties of Probability Distributions • B.4.1 Mean, median and mode • For a discrete random variable the expected value is: Where f is the discrete PDF of x Principles of Econometrics, 3rd Edition

  3. B.4.1 Mean, median and mode • For a continuous random variable the expected value is: The mean has a flaw as a measure of the center of a probability distribution in that it can be pulled by extreme values. Principles of Econometrics, 3rd Edition

  4. B.4.1 Mean, median and mode • For a continuous distribution the median of X is the value m such that • In symmetric distributions, like the familiar “bell-shaped curve” of the normal distribution, the mean and median are equal. • The mode is the value of X at which the pdf is highest. Principles of Econometrics, 3rd Edition

  5. B.4.2 Expected values of functions of a random variable Where g is any function of x, in particular; Principles of Econometrics, 3rd Edition

  6. B.4.2 Expected values of functions of a random variable Principles of Econometrics, 3rd Edition

  7. B.4.2 Expected values of functions of a random variable The variance Principles of Econometrics, 3rd Edition Slide B-7 The variance of a discrete or continuous random variable X is the expected value of

  8. B.4.2 Expected values of functions of a random variable • The variance of a random variable is important in characterizing the scale of measurement, and the spread of the probability distribution. • Algebraically, letting E(X) = μ, Principles of Econometrics, 3rd Edition

  9. B.4.2 Expected values of functions of a random variable Principles of Econometrics, 3rd Edition Slide B-9 The variance of a constant is?

  10. B.4.2 Expected values of functions of a random variable Figure B.3 Distributions with different variances Principles of Econometrics, 3rd Edition

  11. B.4.2 Expected values of functions of a random variable Principles of Econometrics, 3rd Edition

  12. B.4.2 Expected values of functions of a random variable Principles of Econometrics, 3rd Edition

  13. B.4.3 Expected values of several random variables Principles of Econometrics, 3rd Edition

  14. B.4.3 Expected values of several random variables Principles of Econometrics, 3rd Edition

  15. B.4.3 Expected values of several random variables Figure B.4 Correlated data Principles of Econometrics, 3rd Edition

  16. Covariance and correlation coefficient • If X and Y are independent random variables then the covariance and correlation between them are zero. The converse of this relationship is not true. Principles of Econometrics, 3rd Edition

  17. Covariance and correlation coefficient Principles of Econometrics, 3rd Edition Slide B-17 The correlation coefficient is a measure of linear correlation between the variables Its values range from -1 (perfect negative correlation) and 1 (perfect positive correlation)

  18. B.4.3 Expected values of several random variables • If a and b are constants then: Principles of Econometrics, 3rd Edition

  19. B.4.3 Expected values of several random variables So: Why is that? (and of course the same happens for the case of var(X-Y)) Principles of Econometrics, 3rd Edition Slide B-19 • If a and b are constants then:

  20. B.4.3 Expected values of several random variables • If X and Y are independent then: Principles of Econometrics, 3rd Edition

  21. B.4.3 Expected values of several random variables Otherwise this expression would have to include all the doubling of each of the (non-zero) pairwise covariances between variables as summands as well Principles of Econometrics, 3rd Edition Slide B-21 • If X and Y are independent then:

  22. B.4.4 The Simple Experiment Again Principles of Econometrics, 3rd Edition

  23. B.5 Some Important Probability Distributions • B.5.1 The Normal Distribution • If X is a normally distributed random variable with mean μ and variance σ2, it can be symbolized as Principles of Econometrics, 3rd Edition

  24. B.5.1 The Normal Distribution Figure B.5a Normal Probability Density Functions with Means μ and Variance 1 Principles of Econometrics, 3rd Edition

  25. B.5.1 The Normal Distribution Figure B.5b Normal Probability Density Functions with Mean 0 and Variance σ2 Principles of Econometrics, 3rd Edition

  26. B.5.1 The Normal Distribution • A standard normal random variable is one that has a normal probability density function with mean 0 and variance 1. • The cdf for the standardized normal variable Z is Principles of Econometrics, 3rd Edition

  27. B.5.1 The Normal Distribution Principles of Econometrics, 3rd Edition

  28. B.5.1 The Normal Distribution • A weighted sum of normal random variables has a normal distribution. Principles of Econometrics, 3rd Edition

  29. B.5.2 The Chi-square Distribution Principles of Econometrics, 3rd Edition

  30. B.5.2 The Chi-square Distribution Figure B.6 The chi-square distribution Principles of Econometrics, 3rd Edition

  31. B.5.3 The t-Distribution • A “t” random variable (no upper case) is formed by dividing a standard normal random variable by the square root of an independent chi-square random variable, , that has been divided by its degrees of freedom m. Principles of Econometrics, 3rd Edition

  32. B.5.3 The t-Distribution Figure B.7 The standard normal and t(3) probability density functions Principles of Econometrics, 3rd Edition

  33. B.5.4 The F-Distribution • An F random variable is formed by the ratio of two independent chi-square random variables that have been divided by their degrees of freedom. Principles of Econometrics, 3rd Edition

  34. B.5.4 The F-Distribution Figure B.8 The probability density function of an F random variable Principles of Econometrics, 3rd Edition

  35. Keywords • binary variable • binomial random variable • cdf • chi-square distribution • conditional pdf • conditional probability • continuous random variable • correlation • covariance • cumulative distribution function • degrees of freedom • discrete random variable • expected value • experiment • F-distribution • joint probability density function • marginal distribution • mean • median • mode • normal distribution • pdf • probability • probability density function • random variable • standard deviation • standard normal distribution • statistical independence • variance Principles of Econometrics, 3rd Edition

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