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ECON 6002 Econometrics Memorial University of Newfoundland Review of Probability Concepts Adapted from Vera Tabakova’s notes 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 Slide B-2 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 Slide B-3 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. f ( x) P X x f ( x1 ) f ( x2 ) Principles of Econometrics, 3rd Edition f ( xn ) 1 Slide B-4 Principles of Econometrics, 3rd Edition Slide B-5 Figure B.1 College Employment Probabilities Principles of Econometrics, 3rd Edition Slide B-6 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 X is less than or equal to a specific value x F x P X x Principles of Econometrics, 3rd Edition Slide B-7 Principles of Econometrics, 3rd Edition Slide B-8 For example, a binomial random variable X is the number of successes in n independent trials of identical experiments with probability of success p. n x P X x f x p (1 p ) n x x n n! where n! n (n 1 n 2 x x! n x ! Principles of Econometrics, 3rd Edition (B.1) 21 Slide B-9 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 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 First: for winning only once in three weeks, likelihood is 0.189, see? n! x! nx ! 3! 1! 31 ! 3 Times px 1 pnx 0. 7 1 1 0. 72 0. 063 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… The likelihood of winning exactly 2 games, no more or less: n! 3! 3! 321 3 211 x! nx ! 2! 32 ! 2! 1 ! px 1 pnx 0. 7 2 1 0. 71 0. 147 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 So 3 times 0.147 = 0.441 is the likelihood of winning exactly 2 games 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 And 0.343 is the likelihood of winning exactly 3 games n! x! nx ! 3! 3! 33 ! 3! 3!1 3! 3!1 1 px 1 pnx 0. 7 3 1 0. 70 0. 343 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 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) 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 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 So we were looking for 1- binomial(3,2,0.7) Principles of Econometrics, 3rd Edition Slide B-16 Figure B.2 PDF of a continuous random variable Principles of Econometrics, 3rd Edition Slide B-17 P 20 X 40 40 f x dx .355 20 P X x x f t dt F x P 20 X 40 F (40) F (20) .649 .294 .355 Principles of Econometrics, 3rd Edition Slide B-18 1 2 X 3 4 high school diploma or less some college four year college degree advanced degree 0 if had no money earnings in 2002 Y 1 if had positive money earnings in 2002 Principles of Econometrics, 3rd Edition Slide B-19 f x, y 1 x Principles of Econometrics, 3rd Edition y Slide B-20 f X ( x ) f ( x, y ) for each value X can take y f Y ( y ) f ( x, y ) (B.2) for each value Y can take x 4 f Y y f x, y y 0,1 x 1 fY 1 .19 .06 .04 .02 .31 Principles of Econometrics, 3rd Edition Slide B-21 Principles of Econometrics, 3rd Edition Slide B-22 P(Y y, X x) f ( x, y ) f ( y | x) P(Y y | X x) P X x f X ( x) Principles of Econometrics, 3rd Edition y f y | X 3 0 .04/.18=.22 1 .14/.18=.78 (B.3) Slide B-23 Two random variables are statistically independent if the conditional probability that Y = y given that X = x, is the same as the unconditional probability that Y = y. P Y y | X x P Y y (B.4) f ( x, y) f ( y | x) fY ( y ) f X ( x) (B.5) f ( x, y ) f X ( x ) f Y ( y ) (B.6) Principles of Econometrics, 3rd Edition Slide B-24 Y = 1 if shaded Y = 0 if clear X = numerical value (1, 2, 3, or 4) Principles of Econometrics, 3rd Edition Slide B-25 Principles of Econometrics, 3rd Edition Slide B-26 Principles of Econometrics, 3rd Edition Slide B-27 Principles of Econometrics, 3rd Edition Slide B-28 B.4.1 Mean, median and mode E[ X ] x1P X x1 x2 P X x2 xn P X xn (B.7) For a discrete random variable the expected value is: E[ X ] x1 f ( x1 ) x2 f ( x2 ) xn f ( xn ) n xi f ( xi ) xf ( x) i 1 (B.8) x Where f is the discrete PDF of x Principles of Econometrics, 3rd Edition Slide B-29 For a continuous random variable the expected value is: EX xf x dx 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 Slide B-30 For a continuous distribution the median of X is the value m such that P X m P( X m) .5 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 Slide B-31 E[ g ( X )] g ( x) f ( x) (B.9) x Where g is any function of x, in particular; E aX aE X (B.10) E g X g x f x axf x a xf x aE X Principles of Econometrics, 3rd Edition Slide B-32 E aX b aE X b E g1 X g2 X E g1 X E g2 X Principles of Econometrics, 3rd Edition (B.11) (B.12) Slide B-33 The variance The variance of a discrete or continuous random variable X is the expected value of g X X E X Principles of Econometrics, 3rd Edition 2 Slide B-34 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, var( X ) E X E[ X 2 ] 2 2 Principles of Econometrics, 3rd Edition 2 (B.13) Slide B-35 The variance of a constant is? Principles of Econometrics, 3rd Edition Slide B-36 Figure B.3 Distributions with different variances Principles of Econometrics, 3rd Edition Slide B-37 var(aX b) a2 var( X ) (B.14) var(aX b) E aX b E aX b E aX b a b 2 2 E a X a E X a 2 var X 2 Principles of Econometrics, 3rd Edition 2 2 Slide B-38 3 E X skewness 3 4 E X kurtosis 4 Principles of Econometrics, 3rd Edition Slide B-39 E[ g ( X , Y )] g ( x, y ) f ( x, y ) x (B.15) y E X Y E ( X ) E (Y ) (B.16) E X Y x y f x, y xf x, y yf x, y x y x y x y x f x, y y f x, y xf x yf y x y y x x y E X E Y Principles of Econometrics, 3rd Edition Slide B-40 E (aX bY c) aE ( X ) bE (Y ) c (B.17) E XY E g X , Y xyf x, y xyf x f y x y x y xf x yf y E X E Y if X and Y are independent. x y g ( X , Y ) ( X X )(Y Y ) Principles of Econometrics, 3rd Edition (B.18) Slide B-41 Figure B.4 Correlated data Principles of Econometrics, 3rd Edition Slide B-42 Covariance and correlation coefficient cov( X ,Y ) XY E X X Y Y E XY X Y cov X , Y XY var( X ) var(Y ) X Y (B.19) (B.20) 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 Slide B-43 Covariance and correlation coefficient cov X , Y XY var( X ) var(Y ) X Y (B.20) 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) Principles of Econometrics, 3rd Edition Slide B-44 If a and b are constants then: var aX bY a 2 var( X ) b2 var(Y ) 2ab cov( X ,Y ) (B.21) var X Y var( X ) var(Y ) 2cov( X , Y ) (B.22) var X Y var( X ) var(Y ) 2cov( X , Y ) (B.23) Principles of Econometrics, 3rd Edition Slide B-45 If a and b are constants then: var X Y var( X ) var(Y ) 2cov( X , Y ) (B.22) So: var X Yvar Xvar Y2 x y Why is that? (and of course the same happens for the case of var(X-Y)) Principles of Econometrics, 3rd Edition Slide B-46 If X and Y are independent then: var aX bY a 2 var( X ) b2 var(Y ) (B.24) var X Y var( X ) var(Y ) (B.25) var X Y Z var X var Y var Z Principles of Econometrics, 3rd Edition Slide B-47 If X and Y are independent then: var X Y Z var X var Y var Z 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-48 4 E X xf x 1 .1 2 .2 3 .3 4 .4 3 X x 1 E X X 2 X 2 2 2 2 2 1 3 .1 2 3 .2 3 3 .3 4 3 .4 4 .1 1 .2 0 .3 1 .4 1 Principles of Econometrics, 3rd Edition Slide B-49 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 X ~ N , 2 . ( x )2 f ( x) exp , 2 22 2 1 Principles of Econometrics, 3rd Edition x (B.26) Slide B-50 Figure B.5a Normal Probability Density Functions with Means μ and Variance 1 Principles of Econometrics, 3rd Edition Slide B-51 Figure B.5b Normal Probability Density Functions with Mean 0 and Variance σ2 Principles of Econometrics, 3rd Edition Slide B-52 A standard normal random variable is one that has a normal probability density function with mean 0 and variance 1. X Z ~ N (0,1) (B.27) The cdf for the standardized normal variable Z is ( z ) P Z z . Principles of Econometrics, 3rd Edition Slide B-53 a X a a P[ X a] P P Z (B.28) a X a a P[ X a] P P Z 1 (B.29) b a b a P[a X b] P Z (B.30) Principles of Econometrics, 3rd Edition Slide B-54 A weighted sum of normal random variables has a normal distribution. X 1 ~ N 1 , 12 X 2 ~ N 2 , 22 Y a1 X1 a2 X 2 ~ N Y a11 a22 , Y2 a1212 a2222 2a1a212 (B.27) Principles of Econometrics, 3rd Edition Slide B-55 V Z12 Z22 Zm2 ~ (2m) (B32) E[V ] E (2m ) m 2 var[V ] var ( m ) 2m Principles of Econometrics, 3rd Edition (B.33) Slide B-56 Figure B.6 The chi-square distribution Principles of Econometrics, 3rd Edition Slide B-57 A “t” random variable (no upper case) is formed by dividing a standard normal random variable Z ~ N 0,1 by the square root of an independent chi-square random variable, V ~ (2m) , that has been divided by its degrees of freedom m. Z t V Principles of Econometrics, 3rd Edition ~ t( m ) (B.34) m Slide B-58 Figure B.7 The standard normal and t(3) probability density functions Principles of Econometrics, 3rd Edition Slide B-59 An F random variable is formed by the ratio of two independent chisquare random variables that have been divided by their degrees of freedom. V1 m1 F ~ F( m1 ,m2 ) V2 m2 Principles of Econometrics, 3rd Edition (B.35) Slide B-60 Figure B.8 The probability density function of an F random variable Principles of Econometrics, 3rd Edition Slide B-61 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 Principles of Econometrics, 3rd Edition 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 Slide B-62