Econ 325: Introduction to Empirical Economics

Size: px

Start display at page:

Download "Econ 325: Introduction to Empirical Economics"

Elwin Atkinson
5 years ago
Views:

1 Eco 35: Itroductio to Empirical Ecoomics Lecture 3 Discrete Radom Variables ad Probability Distributios Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-1

2 4.1 Itroductio to Probability Distributios Radom Variable Represets a possible umerical value from a radom experimet Radom Variables Discrete Radom Variable Cotiuous Radom Variable Ch. 4 Ch. 5 Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-

3 Discrete Radom Variables Ca oly take o a coutable umber of values Examples: Roll a die twice Let X be the umber of times 4 comes up (the X could be 0, 1, or times) Toss a coi 3 times. Let X be the umber of heads (the X 0, 1,, or 3) Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-3

4 4. Discrete Probability Distributio Experimet: Toss Cois. Let X # heads. Show P(x), i.e., P(X x), for all values of x: 4 possible outcomes T T T H H T H H Probability Distributio x Value Probability 0 1/4.5 1 /4.50 1/4.5 Probability x Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-4

5 Radom variable S {TT, TH, HT, TH} Defie a fuctio X(s) by X({TT})0, X({TH})1, X({HT})1, X({HH}) P(X0) P({TT}) 1/4 P(X1) P ({TH,HT}) 1/ Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-5

6 Defiitio: Radom variable A radom variable X is a fuctio which maps the outcome of a experimet s to the real umber x. X: S the space of X The space of X is give by S X {x: X s x, s S} Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-6

7 Discrete Probability Distributio The space of X {0,1,}. Defie a set A {0,1} i the space of X. The, P(X Î A) å P(X x) P(X 0) + P(X 1) xîa Notatio: Uppercase ``X represets a radom variable ad lowercase ``x represets some costat (e.g., realized value). Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-7

8 4.3 Defiitio: Probability mass fuctio The probability mass fuctio (pmf) of a discrete radom variable X is a fuctio that satisfies the followig properties: 1). 0 f X x 1 ). f X x x < 1 3). P X A f X x x A Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-8

9 Probability mass fuctio f ( x) P(X X x) Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-9

10 Cumulative Probability Fuctio The cumulative probability fuctio, deoted F(x 0 ), is a fuctio defied by the probability of X beig less tha or equal to x 0 å F(x0 ) P(X x 0) fx (x) x x 0 Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-10

11 Questio Defie X # of heads whe you toss cois. What is the probability mass fuctio ad the cumulative distributio fuctio of X? Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-11

12 Questio Defie X a umber you get from rollig a die. What is the probability mass fuctio ad the cumulative distributio fuctio of X? Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-1

13 Expected Value Expected Value (or mea) of a discrete distributio µ E(X) å x f X x (x) Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-13

14 Example What is the expected value whe you roll a die oce? f X i P X i 1 6 for i 1,,, 6 6 E X i 1 il Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-14

15 Clicker Questio 3-1 Defie X # of heads whe you toss cois. What is the expected value of X? A). 0.5 B). 1 C). 1.5 Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-15

16 Variace ad Stadard Deviatio Variace of a discrete radom variable X σ E(X - µ) å(x - x µ) f X (x) Stadard Deviatio of a discrete radom variable X σ σ å(x - x µ) f X (x) Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-16

17 Stadard Deviatio Example Example: Toss cois, X # heads, compute stadard deviatio (recall E(x) 1) σ å(x - x µ) f X (x) σ (0-1) (.5) + (1-1) (.50) + (-1) (.5) Possible umber of heads 0, 1, or Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-17

18 Clicker Questio 3- Toss 1 coi. Let X 1 if it is head ad X0 if it is tail. What is the variace of this radom variable? A). 1 B). 0.5 C). 0.5 D). 0.1 Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-18

19 Fuctios of Radom Variables If P(x) is the probability fuctio of a discrete radom variable X, ad g(x) is some fuctio of X, the the expected value of fuctio g is E[g(X)] å x g(x)f X (x) Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-19

20 Clicker Questio 3-3 Toss 1 coi. Let X 1 if it is head ad X0 if it is tail. Cosider a fuctio g(x) such that g(1) 100 ad g(0) 0. What is E[g(X)]? A). 0 B). 100 C). 50 D). 10 Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-0

21 Liear Fuctios of Radom Variables Let a ad b be ay costats. a) E(a) a ad Var(a) 0 i.e., if a radom variable always takes the value a, it will have mea a ad variace 0 b) E(bX) be(x) ad Var(bX) b Var(X) i.e., the expected value of b X is b E(X) Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-1

22 Liear Fuctios of Radom Variables (cotiued) Let radom variable X have mea µ x ad variace σ x Let a ad b be ay costats. Let Y a + bx The the mea ad variace of Y are E(Y) E(a + bx) a + be(x) Var(Y) Var(a + bx) b Var(X) so that the stadard deviatio of Y is σ Y bσ X Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-

23 Beroulli Distributio Cosider oly two outcomes: success or failure Let p deote the probability of success Let 1 p be the probability of failure Defie radom variable X: X 1 if success, X 0 if failure The the Beroulli probability fuctio is P(X 0) (1- p) ad P(X 1) p Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-3

24 Possible Beroulli Distributio Settigs A survey resposes of ``I will vote for Clito or ``I will vote for Trump A maufacturig plat labels items as either defective or acceptable A marketig research firm receives survey resposes of yes I will buy or o I will ot Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-4

25 Beroulli Distributio Mea ad Variace The mea is µ p µ E(X) xp(x x) (0)(1 p)+ (1)p p x0,1 The variace is σ p(1 p) x0,1 σ E[(X µ) ] (x µ) P(X x) (0 p) (1 p)+ (1 p) p p(1 p) Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-5

26 Questio Let X M, X N, ad X O are three idepedet Beroulli radom variables with P X Q 1 p for i 1,. What is the probability mass fuctio of Y X M + X N + X O? Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-6

27 Questio Let X Q for i 1,,, are idepedet Beroulli radom variables with P X Q 1 p. What is the probability mass fuctio of Y X QLM X Q? Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-7

28 Biomial Distributio Cosider the sum of idepedet Beroulli radom variables: Y å i 1 X i, where P(Xi 0) (1- p) ad P(Xi 1) p P(Yy) probability of y successes i trials, with probability of success p o each trial y umber of successes i sample (y 0, 1,,..., ) p sample size (umber of trials or observatios) probability of success Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-8

29 Probability mass fuctio of Biomial distributio P(Yy)! y! ( - y )! y - y p (1- p) P(y) probability of y successes i trials, with probability of success p o each trial y umber of successes i sample, (y 0, 1,,..., ) sample size (umber of trials or observatios) p probability of success Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-9

30 Example: Calculatig a Biomial Probability Flip 3 cois ad let Y be the umber of heads. What is the probability that Y1? Y 1, 3, ad p 0.5 P(Y 1)! y!( - 3! (0.5) 1!(3-1)! (3)(0.5)(0.5) 3/ 8 p y)! y (1- p) 1 - y (1-0.5) 3-1 Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-30

31 Clicker Questio 3-4 The Miesota Twis are to play a series of 5 games agaist the Red Sox. For ay oe game, the probability of a Twi s wi is 0.5. The outcome of the 5 games are idepedet of oe aother. What is the probability that Twis wi all 5 games? A). 1/ B). 1/4 C). 1/16 D).1/3 Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-31

32 Biomial Distributio Mea ad Variace Mea µ E(Y) E( å X i ) å E(X i) i 1 i 1 p Variace ad Stadard Deviatio σ p(1- p) σ p(1- p) Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-3

33 Average of idepedet Beroulli radom variable Cosider the sample average of idepedet Beroulli radom variable: X 1 å i 1 X i, with P(Xi 1) p ad P(Xi 0) (1- p) X The, XY is related to Biomial radom variable Y QLM X Q as X 1 å i 1 X i 1 Y Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-33

34 Clicker Questio 3-5 Cosider the sample average of idepedet Beroulli radom variable: X 1 å i 1 X i, with P(Xi 1) p ad P(Xi 0) (1- p) What is E(XY)? A). p B). 1-p C). p Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-34

35 4.7 Joit probability mass fuctios A joit probability mass fuctio is used to express the probability that X takes the specific value x ad simultaeously Y takes the value y, as a fuctio of x ad y f(x, y) P(X x,y y) The margial probabilities are f å X (x) f(x, y) y å fy (y) f(x, y) x Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-35

36 Stochastic Idepedece The joitly distributed radom variables X ad Y are said to be idepedet if ad oly if f(x, y) f (x)f (y) for all possible pairs of values x ad y X Y A set of k radom variables are idepedet if ad oly if f(x, x,!, x ) f X (x1)f X (x )! f 1 k X 1 k (x k ) Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-36

37 Clicker Questio 3-6 Is X ad Y stochastically idepedet? Y30 Y60 Y100 Margial Dist. of X X X Margial Dist. of Y A). X ad Y are idepedet B). X ad Y are ot idepedet Ch Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall

38 Coditioal probability mass fuctios The coditioal probability mass fuctio of the radom variable Y is defie by fy X (y x) f(x, y) f (x) X Similarly, the coditioal probability mass fuctio of X give Y y is: fx Y (x y) f(x, y) f (y) Y Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-38

39 Questio What is the coditioal mass probability mass fuctio of Y give X1? Y30 Y60 Y100 Margial Dist. of X X X Margial Dist. of Y Ch Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall

40 Coditioal Mea ad Variace The coditioal mea is µ Y X x E Y X [Y X y f(y E[Y Xx] is a fuctio of x ad, therefore, is also called as ``coditioal expectatio fuctio (CEF) x] å y x) The coditioal variace is σ å Y X x EY X[(Y - µ Y X x ) X x] (y - µ Y X x ) f(y x) y Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-40

41 Clicker Questio 3-7 What is E[X Y30]? Y30 Y60 Y100 Margial Prob of X X X Margial Prob of Y A). 1/ B). 1/3 C). /3 D).1/4 Ch Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall

42 Clicker Questio 3-8 What is Var[X Y30]? Y30 Y60 Y100 Margial Prob of X X X Margial Prob of Y A). 1/9 B). 1/3 C). /9 D). /7 Ch. 3-4 Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall

43 Clicker Questio 3-9 Suppose X ad Y are stochastically idepedet. The, A). the coditioal mea of X give Yy is the same as the ucoditioal mea of X. B). the coditioal mea of X give Yy may ot be the same as the ucoditioal mea of X. Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-43

44 E ] [Y X] as a radom variable Viewig X as a radom variable, E ] [Y X] is a radom variable because the value of E ] [Y X] depeds o a realizatio of X. The Law of Iterated Expectatios: E [E ] [Y X]] E ] [Y] Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-44

45 Covariace Let X ad Y be discrete radom variables with meas μ X ad μ Y The expected value of (X - μ X )(Y - μ Y ) is called the covariace betwee X ad Y For discrete radom variables Cov(X, Y) E[(X åå(x - µ X )(y - - µ X )(Y - µ Y )] µ Y )f(x, y) x y Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-45

46 Correlatio The correlatio betwee X ad Y is: ρ Corr(X,Y) Cov(X, Y) σ X σ Y ρ 0 : o liear relatioship betwee X ad Y ρ > 0 : positive liear relatioship betwee X ad Y whe X is high (low) the Y is likely to be high (low) ρ +1 : perfect positive liear depedecy ρ < 0 : egative liear relatioship betwee X ad Y whe X is high (low) the Y is likely to be low (high) ρ -1 : perfect egative liear depedecy Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-46

47 Ucorrelatedess, Mea Idepedece, Stochastic Idepedece X ad Y are said to be ucorrelated whe Cov(X,Y)0 or ρ 0. X is said to be mea idepedet of Y whe E ] X Y E [X]. X ad Y are said to be stochastically idepedet whe f x, y f x f ] y. Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-47

48 Ucorrelatedess, Mea Idepedece, Stochastic Idepedece Stochastic Idepedece f x, y f x f ] y Mea Idepedece E ] X Y E [X] or E ] Y X E ] [Y] Ucorrelatedess Cov(X,Y)0 Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-48

49 Var ax + by For ay costat a ad b ad ay two radom variables X ad Y, Var ax + by a N Var X + b N Var Y +ab Cov X, Y Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-49

50 Clicker Questio 3-10 Which of the followig is true. A). Var ax + by a N Var X + b N Var Y +ab Corr X, Y B). Var ax + by a N Var X + b N Var Y +abσ σ ] Corr X, Y C). Var ax + by a N Var X + b N Var Y +abcorr X, Y /σ σ ] Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-50

51 Portfolio Aalysis Let radom variable X be the share price for stock A Let radom variable Y be the share price for stock B The market value, W, for the portfolio is give by the liear fuctio W ax + by ``a ad ``b are the umbers of shares of stock A ad B, respectively. The retur from holdig the portfolio W: DW adx + bdy Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-51

52 Portfolio Aalysis (cotiued) The mea value for W is The variace for W is E[ DW] E[aDX + bdy] ae[ DX] + be[ DY] σ DW a σd X + b σd or usig the correlatio formula Y + abcov( DX, DY) σ DW a σ DX + b σ DY + abcorr( DX, DY)σ DX σ DY Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-5

53 Example: Ivestmet Returs Retur per $100 for two types of ivestmets Ivestmet P( X, Y) Ecoomic coditio Bod Fud X Aggressive Fud Y 0. Recessio + $ 7 - $0 0.5 Stable Ecoomy Expadig Ecoomy E( X) (7)(.) +(4)(.5) + ()(.3) 4 E( Y) (-0)(.) +(6)(.5) + (35)(.3) 9.5 Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-53

54 Computig the Stadard Deviatio for Ivestmet Returs Ivestmet P( X, Y) Ecoomic coditio Bod Fud X Aggressive Fud Y 0. Recessio + $ 7 - $0 0.5 Stable Ecoomy Expadig Ecoomy s D X Var( DX) (7-4) (0.) + (4-4) (0.5) + ( - 4) (0.3) 3» 1.73 s D Y Var( DY) (-0-9.5) (0.) + (6-9.5) (0.5) + (35-9.5) (0.3) 375.5» Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-54

55 Covariace for Ivestmet Returs Ivestmet P( X, Y) Ecoomic coditio Bod Fud X Aggressive Fud Y 0. Recessio + $ 7 - $0 0.5 Stable Ecoomy Expadig Ecoomy s D XDY Cov( DX, DY) (7-4)(-0-9.5)(.) + + ( - 4)(35-9.5)(.3) (4-4)(6-9.5)(.5) -33 Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-55

56 Portfolio Example Ivestmet X: E( X) 4 σ X 1.73 Ivestmet Y: E( Y) 9.5 σ Y 19.3 σ X Y -33 Suppose 40% of the portfolio (W) is i Ivestmet X ad 60% is i Ivestmet Y: E( DW).4(4) + (.6)(9.5) 7.3 Var( DW) (.4) (1.73) + (.6) (19.3) + (.4)(.6)(-33) The portfolio retur ad portfolio variability are betwee the values for ivestmets X ad Y cosidered idividually Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-56

57 Iterpretig the Results for Ivestmet Returs The aggressive fud has a higher expected retur, but much more risk E( Y) 9.5 > E( X) 4 but σ Y 19.3 > σ X 1.73 The Covariace of -33 idicates that the two ivestmets are egatively related ad will vary i the opposite directio Copyright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 4-57

Some Basic Probability Concepts. 2.1 Experiments, Outcomes and Random Variables

Some Basic Probability Concepts. 2.1 Experiments, Outcomes and Random Variables Some Basic Probability Cocepts 2. Experimets, Outcomes ad Radom Variables A radom variable is a variable whose value is ukow util it is observed. The value of a radom variable results from a experimet;