Review of Probability Concepts 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: (B.7) (B.8) Where f is the discrete PDF of x Principles of Econometrics, 3rd Edition

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

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

B.4.2 Expected values of functions of a random variable Where g is any function of x, in particular; (B.10) Principles of Econometrics, 3rd Edition

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

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

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, (B.13) Principles of Econometrics, 3rd Edition

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

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

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

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

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

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

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

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. (B.19) (B.20) Principles of Econometrics, 3rd Edition

Covariance and correlation coefficient 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) (B.20) Principles of Econometrics, 3rd Edition Slide B-17 17

B.4.3 Expected values of several random variables If a and b are constants then: (B.21) (B.22) (B.23) Principles of Econometrics, 3rd Edition

B.4.3 Expected values of several random variables If a and b are constants then: (B.22) 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 19

B.4.3 Expected values of several random variables If X and Y are independent then: (B.24) (B.25) Principles of Econometrics, 3rd Edition

B.4.3 Expected values of several random variables If X and Y are independent then: 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 21

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

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 (B.26) Principles of Econometrics, 3rd Edition

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

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

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 (B.27) Principles of Econometrics, 3rd Edition

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

B.5.1 The Normal Distribution A weighted sum of normal random variables has a normal distribution. (B.27) Principles of Econometrics, 3rd Edition

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

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

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. (B.34) Principles of Econometrics, 3rd Edition

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

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. (B.35) Principles of Econometrics, 3rd Edition

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

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