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### Fec512.02

1. 1. Random Variables and Summary Measures Istanbul Bilgi University FEC 512 Financial Econometrics-I Asst. Prof. Dr. Orhan Erdem
2. 2. Introduction to Probability Distributions Random Variable Represents a numerical value from a random event Random Variables Discrete Continuous Random Variable Random Variable Lecture 2-2 FEC 512 Probability Distributions
3. 3. Definitions The r.v. is discrete if it takes countable number of values. The discrete r.v. X has probability density function (pdf) f:R→[0,1] fiven by f(x)=P(X=x) The r.v. is continuous if its takes uncountable number of values. Lecture 2-3 FEC 512 Probability Distributions
4. 4. Examples Stock prices are discrete random variables, because they can only take on certain values, such as 10.00TL, 10.01TL and 10.02TL and not 10.005TL, since stocks have a minimum tick size of 0.01TL. By way of contrast, stock returns are continuous not discrete random variables, since a stock's return could be any number. Lecture 2-4 FEC 512 Probability Distributions
5. 5. Dicrete Random Variables: Examples Roll a die twice: Let x be the number of times 4 comes up (then x could be 0, 1, or 2 times) Toss a coin 5 times. Let x be the number of heads (then x = 0, 1, 2, 3, 4, or 5) Lecture 2-5 FEC 512 Probability Distributions
6. 6. Discrete Probability Distribution Experiment: Toss 2 Coins. Let x = # heads. 4 possible outcomes Probability Distribution T T x Value Probability 0 1/4 = .25 T H 1 2/4 = .50 2 1/4 = .25 H T Probability .50 H .25 H 0 1 2 x Lecture 2-6 FEC 512 Probability Distributions
7. 7. Discrete Probability Distribution Function (P.d.f.) 0 ≤ P(xi) ≤ 1 for each xi Σ P(xi) = 1 Lecture 2-7 FEC 512 Probability Distributions
8. 8. Cumulative Distribution Function(c.d.f.) Cumulative distribution function of X is FX(x)=P(X≤x) If X has a pdf then u=x ∑f FX ( x) = u u = −∞ Example: Draw the c.d.f of the prev. example Lecture 2-8 FEC 512 Probability Distributions
9. 9. Summary Measures: Location Expected Value of a discrete distribution (Weighted Average) E(x) = Σxi P(xi) x P(x) Example: Toss 2 coins, 0 .25 x = # of heads, 1 .50 compute expected value of x: 2 .25 E(x) = (0 x .25) + (1 x .50) + (2 x .25)=1.0 Lecture 2-9 FEC 512 Probability Distributions
10. 10. The Allais Example-1 1 Lottery Probability 1 Outcome 500,000 2.Lottery Probability 0.10 0.89 0.01 Outcome 2,500,000 500,000 0 Which one do you prefer? It is common for ind. to express 1.Lottery is better than 2.Lottery Lecture 2-10 FEC 512 Probability Distributions
11. 11. The Allais Example-2 1 Lottery Probability 0 0,11 0.89 Outcome 2,500,00 500,000 0 2.Lottery Probability 0,10 0 0.90 Outcome 2,500,00 500,000 0 Which one do you prefer? It is common for ind. to express 2..Lottery is better than 1.Lottery Lecture 2-11 FEC 512 Probability Distributions
12. 12. Summary Measures: Dispersion Standard Deviation of a discrete distribution n ∑{x σx = − E(x)} P(x i ) 2 i i=1 where: E(x) = Expected value of the random variable P(x) = Probability of the random variable having the value of x Lecture 2-12 FEC 512 Probability Distributions
13. 13. Summary Measures: Dispersion Example: Toss 2 coins, x = # heads, compute standard deviation (recall E(x) = 1) n ∑{x σx = − E(x)} P(x i ) 2 i i=1 σ x = (0 − 1)2 (.25) + (1 − 1)2 (.50) + (2 − 1)2 (.25) = .50 = .707 Possible number of heads = 0, 1, or 2 Lecture 2-13 FEC 512 Probability Distributions
14. 14. Example: Random Walk Assume that at each time step the price can either increase or decrease by a fixed amount ∆>0. Suppose that P1: is the probability of an increase (0< P1 <1) P2 : is the probability of a decrease. (0< P2 <1) Random Variable: If X is the change in a single step, then the set of possible values of X is {x1= ∆, x2=- ∆} and their probabilities are {P1, P2} What is the exp. value of a random walk if P1=P2 =0.5? Lecture 2-14 FEC 512 Probability Distributions
15. 15. Chebyshev’s Inequality Let X be a r.v. with expected value µ and finite variance σ2. Then for any real number m> 0, 1 P( X − µ ≥ mσ ) ≤ 2 m or 1 P( X − µ ≤ mσ ) ≤ 1 − 2 m Lecture 2-15 FEC 512 Probability Distributions
16. 16. Two Discrete Random Variables Expected value of the sum of two discrete random variables: E(x + y) = E(x) + E(y) = Σ x P(x) + Σ y P(y) Lecture 2-16 FEC 512 Probability Distributions
17. 17. Conditional Expectation The conditional pdf of Y given X=x written fYlX(y l x)=P(Y=ylX=x). E(Y l X=x) is called conditional expectation of Y given X, defined as E(Y l X)=ΣyfYlX(y l x) Although conditional expectation sounds like a number it is actually a r.v. Lecture 2-17 FEC 512 Probability Distributions
18. 18. Bivariate Distributions Situations where we are interested at the same time in a pair of r.v. Defined over a joint sample space. If X and Y are disrete r.v., we write the prob that X will take on the value x and Y will take on the value y as P(X=x,Y=y), the joint pdf. If X and Y are cont r.v. the joint pdf of X,Y is the function fX,Y(x,y) which display the joint distribution of X,Y. Lecture 2-18 FEC 512 Probability Distributions
19. 19. Example Determine the value of k for which the function given by f(x,y)=kxy for x=1,2,3; y=1,2,3 can serve as a joint pdf. Solution: Substituting values of x,y we get f(1,1)=k; f(1,2)=2k; …f(3,3)=9k k+2k+3k+2k+4k+6k+3k+6k+9k=1 36k=1 and k=1/36 Lecture 2-19 FEC 512 Probability Distributions
20. 20. Example: Conditional Pdf X 0 1 2 0 1/6 1/3 1/12 7/12 Y 1 2/9 1/6 - 7/18 2 1/36 - - 1/36 5/12 ½ 1/12 1 1/ 6 6 P( Y = 0 X = 0) = = 5 / 12 15 2/9 8 P( Y = 1 X = 0) = = 5 / 12 15 1 / 36 1 P( Y = 0 X = 0) = = 5 / 12 15 Lecture 2-20 FEC 512 Probability Distributions
21. 21. Continuous Random Variables has a probability density function (pdf) fX such that (a ) f ( x) ≥ 0 fo r a ll x , +∞ ∫ f ( x ) d x = 1. (b ) -∞ ( c ) F o r a n y a , b , w ith - ∞ < a < b < + ∞ , b ∫ w e have P (a ≤ X ≤ b ) = f ( x)dx a Examples: Changes in stock prices Lecture 2-21 FEC 512 Probability Distributions
22. 22. Cumulative Distribution Function * Cumulative distribution function (CDF) of X is FX ( x ) = P ( X ≤ x ) If X has a pdf then x ∫f FX ( x ) = (u )du X −∞ Lecture 2-22 FEC 512 Probability Distributions
23. 23. Moments Lecture 2-23 FEC 512 Probability Distributions
24. 24. Example: Continuous Probability Distributions Ex. Suppose that X is a continuous random variable with pdf f ( x) = 2 x, 0 < x < 1, = 0, elsewhere. Hence the cdf is given by 1,2 F ( x) = 0, x ≤ 0, if 1 x 0,8 ∫ 2s ds = =x, if 0 < x ≤ 1, 2 F(x) 0,6 0 0,4 = 1, if x > 1. 0,2 0 The graph of F(x) 0 0,2 0,4 0,6 0,8 1 1,2 x Lecture 2-24 FEC 512 Probability Distributions
25. 25. Marginal Distributions X 0 1 2 0 1/6 1/3 1/12 7/12 Y 1 2/9 1/6 - 7/18 2 1/36 - - 1/36 5/12 ½ 1/12 1 Marginal Distribution of Y : 7 7 1 P(Y = 0) = ; P(Y = 1) = ; P(Y = 2) = 12 18 36 Lecture 2-25 FEC 512 Probability Distributions
26. 26. Conditional Distribution and Expectation i. Discrete Case P( A ∩ B) Before we have seen P( A B) = , Similarly P( B) P( X = x Y = y ) P( X = x Y = y ) = is the conditional pdf of X, Y. P (Y = y ) Remember that Y is fixed here. Lecture 2-26 FEC 512 Probability Distributions
27. 27. Conditional Distribution and Expectation ii. Continuous Case (y x ) is given by The conditional pdf, written as f Y X fY X (y x) = f ( x, y ) . f X ( x) ∞ E (Y X ) = ∫ yf ( y x)dy YX −∞ Lecture 2-27 FEC 512 Probability Distributions
28. 28. Martingale Given an information set available at time t, I t , a sequence of r.v. Pt is called a martingale w.r.t info set I t if E[ Pt +1 I t ] = Pt Lecture 2-28 FEC 512 Probability Distributions
29. 29. Some Common Properties
30. 30. Skewness The skewness of a r.v. measures the symmetry of a dist. About its mean value. { } E [ X − E ( X )]3 Skew( X ) = σ x3 n ∑(X − E ( X ))3 P( X i ) i i =1 = if X is discrete. σx 3 ∞ ∫ ( X − E ( X ))3 f x = −∞ if X is continuous σx 3 Lecture 2-30 FEC 512 Probability Distributions
31. 31. Kurtosis The kurtosis of a r.v. measures the thickness in the tails of a distribution. { } E [ X − E ( X )] 4 Kurt ( X ) = σ x4 n ∑ (X − E ( X )) 4 P ( X i ) i i =1 = if X is discrete. σ x4 ∞ ∫ (X − E ( X )) 4 f x = −∞ if X is continuous σx 4 Lecture 2-31 FEC 512 Probability Distributions
32. 32. Example x 0 1 2 P(x) 0.25 0.5 0.25 We know that E(X)=µ=1, σ=0.707 from previous example. Skew(X)=[(0-1)30.25+(1-1)3 0.5+(2-1)30.25] /(0.707)3=0. So it is symmetric. H.W. Find its kurtosis. Lecture 2-32 FEC 512 Probability Distributions
33. 33. Covariance Covariance between two r.v. σ XY = E [{X − E ( X )}{Y − E (Y )}] Lecture 2-33 FEC 512 Probability Distributions
34. 34. Covariance (cont.) If X,Y are discrete r.v: σxy = Σ [xi – E(x)][yj – E(y)]P(xiyj) where: P(xi ,yj) = joint probability of xi and yj. Lecture 2-34 FEC 512 Probability Distributions
35. 35. Useful Formulas Lecture 2-35 FEC 512 Probability Distributions
36. 36. Interpreting Covariance Covariance between two discrete random variables: σxy > 0 x and y tend to move in the same direction σxy < 0 x and y tend to move in opposite directions σxy = 0 x and y do not move closely together Lecture 2-36 FEC 512 Probability Distributions
37. 37. Correlation Coefficient The Correlation Coefficient shows the strength of the linear association between two variables σxy ρ= σx σy where: ρ = correlation coefficient (“rho”) σxy = covariance between x and y σx = standard deviation of variable x σy = standard deviation of variable y Lecture 2-37 FEC 512 Probability Distributions
38. 38. Interpreting the Correlation Coefficient The Correlation Coefficient always falls between -1 and +1 ρ=0 x and y are not linearly related. The farther ρ is from 0, the stronger the linear relationship: ρ = +1 x and y have a perfect positive linear relationship ρ = -1 x and y have a perfect negative linear relationship * A strong nonlinear relationship may or or may not imply a high correlation Lecture 2-38 FEC 512 Probability Distributions
39. 39. Lecture 2-39 FEC 512 Probability Distributions
40. 40. Independence A r.v. X is independent of Y if knowledge about Y does not influence the likelihood that X=x for all possible values of x. and y. (Similarly for Y) Holds for both type of r.v. Lecture 2-40 FEC 512 Probability Distributions
41. 41. Independence The r.v. X and Y are independent if and only if f X,Y (x, y) = f X (x)f Y (y) for cont r.v. or P(X = x andY = y) = P(X = x)P(Y = y) for disc.r.v. for all x, y ∈ R. If X and Y are independent then E(XY) = E(X)E(Y) If E(XY) = E(X)E(Y), then X and Y are uncorrelated Thus independence ⇒ E(XY) = E(X)E(Y) ⇒ uncorrelatedness. The converse is not true. Lecture 2-41 FEC 512 Probability Distributions
42. 42. Linear Functions of a Random Variable Let X be a r.v. Either discrete or cont. E(X)=µ, Var(X)=σ2.Define a new r.v. Y as Y=aX+b. Then E(Y)=aE(X)+b=a µ+b Var(Y)=a2 σ2 Lecture 2-42 FEC 512 Probability Distributions
43. 43. Linear Combinations of Two Random Variables Let X~(µX,σX2 ) and Y~(µY,σY2 ) and σXY=cov(X,Y). If Z=aX+bY where a,b are constants, then Z~(µZ,σZ2 ) where µZ=a µX +b µY σZ2=a2 σX2 +b2 σY2 +2abσXY=a2 σX2 +b2 σY2 +2abσX σYρ Lecture 2-43 FEC 512 Probability Distributions
44. 44. Linear Combinations of N Random Variables Lecture 2-44 FEC 512 Probability Distributions
45. 45. Diversification As long as security returns are not positively correlated, diversification benefits are possible. The smaller the correlation between security returns, the greater the cost of not diversifying. σZ2=a2 σX2 +b2 σY2 +2abσX σYρ Sigma(Z) 3 2.5 2 1.5 1 0.5 0 -1 -0.5 0 0.5 1 Correlation Lecture 2-45 FEC 512 Probability Distributions