Fec512.02

Random Variables and

Summary Measures

Istanbul Bilgi University

FEC 512 Financial Econometrics-I

Asst. Prof. Dr. Orhan Erdem

FEC 512 Probability Distributions Lecture 2-2

Introduction to Probability Distributions

� Random Variable

� Represents a numerical value from a

random event

Random

Variables

Discrete

Random Variable

Continuous

Random Variable


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.


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.


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)


x Value Probability

0 1/4 = .25

1 2/4 = .50

2 1/4 = .25

Experiment: Toss 2 Coins. Let x = # heads.

T

T

Discrete Probability Distribution

4 possible outcomes

T

T

H

H

H H

Probability Distribution

0 1 2 x

.50

.25

Probability


� 0 ≤ P(xi) ≤ 1 for each xi

� Σ P(xi) = 1

Discrete Probability Distribution

Function (P.d.f.)


Cumulative Distribution Function(c.d.f.)

� Cumulative distribution function of X is

FX(x)=P(X≤x)

� If X has a pdf then

� Example: Draw the c.d.f of the prev. example

∑=

−∞=

=xu

u

uX fxF )(


Summary Measures: Location

� Expected Value of a discrete distribution(Weighted Average)

E(x) = Σxi P(xi)

� Example: Toss 2 coins,

x = # of heads,

compute expected value of x:

E(x) = (0 x .25) + (1 x .50) + (2 x .25)=1.0

x P(x)

0 .25

1 .50

2 .25


The Allais Example-1

� 1 Lottery

� 2.Lottery

� Which one do you prefer?

� It is common for ind. to express 1.Lottery is better than2.Lottery

500,000Outcome

1Probability

0500,0002,500,000Outcome

0.010.890.10Probability


The Allais Example-2

� 1 Lottery

� 2.Lottery

� Which one do you prefer? It is common for ind. to express

2..Lottery is better than 1.Lottery

500,000

0,11

02,500,00Outcome

0.890Probability

500,000

0

02,500,00Outcome

0.900,10Probability


� Standard Deviation of a discrete distribution

where:

E(x) = Expected value of the random variable

P(x) = Probability of the random variable having

the value of x

Summary Measures: Dispersion

2

x i i

1

σ {x E(x)} P(x )n

i=

= −∑


� Example: Toss 2 coins, x = # heads, compute standard deviation (recall E(x) = 1)

Summary Measures: Dispersion

2

x i i

1

σ {x E(x)} P(x )n

i=

= −∑.707.50(.25)1)(2(.50)1)(1(.25)1)(0σ 222

x ==−+−+−=

Possible number of heads

= 0, 1, or 2


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?


Chebyshev’s Inequality

� Let X be a r.v. with expected value µ and finite

variance σ2. Then for any real number m> 0,

2

2

11)(

1)(

mmXP

or

mmXP

−≤≤−

≤≥−

σµ

σµ


� 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)

Two Discrete Random Variables


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.


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.


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


Example: Conditional Pdf

Y

X

11/12½5/12

1/36--1/362

7/18-1/62/91

7/121/121/31/60

210

15

1

12/5

36/10)X0YP(

15

8

12/5

9/20)X1YP(

15

6

12/5

6/10)X0YP(

====

====

====


Continuous Random Variables

� has a probability density function (pdf) fX

such

that

� Examples: Changes in stock prices

( ) ( ) 0

( ) ( ) 1.

( ) , , - ,

( ) ( )

+

-

,

b

a

a f x for a ll x

b f x dx

c For any a b w ith a b

w e have P a X b f x dx

∞

∞

≥

=

∞ < < < +∞

≤ ≤ =

∫

∫


Cumulative Distribution Function

∫∞−

=

≤=

x

XX

X

duufxF

xXPxF

)()(

then pdf a has X If

)()(

is X of (CDF)function on distributi Cumulative*


Moments


Example: Continuous Probability

Distributions

Ex. Suppose that X is a continuous random variable with pdf

Hence the cdf is given by

The graph of F(x) �0

0,2

0,4

0,6

0,8

1

1,2

0 0,2 0,4 0,6 0,8 1 1,2

x

F(x)

( ) 2 , 0 1,

0, .

f x x x

elsewhere

= < <

=

2

0

( ) 0, 0,

2 , 0 1,

1, 1.

x

F x if x

s ds x if x

if x

= ≤

= = < ≤

= >

∫


Marginal Distributions

Y

X

11/12½5/12

1/36--1/362

7/18-1/62/91

7/121/121/31/60

210

36

12)P(Y ;

18

71)P(Y ;

12

70)P(Y

:Y ofon Distributi Marginal

======


Conditional Distribution and Expectation

i. Discrete Case

here. fixed is Yhat Remember t

Y.X, of pdf lconditiona theis )(

)()(

Similarly,)(

)()(seen have weBefore

yYP

yYxXPyYxXP

BP

BAPBAP

=

=====

∩=


Conditional Distribution and Expectation

ii. Continuous Case

( )

( )

( ) ∫∞

∞−

=

=

dyxyyfXYE

xf

yxfxyf

xyf

XY

X

XY

XY

)(

.)(

),(

bygiven is aswritten pdf, lconditiona The


Martingale

ttt PIPE =+ ][

if Iset info w.r.t martingale a called is P r.v. of sequence a

,I t,at time availableset n informatioan Given

1

tt

t

Some Common Properties


Skewness

� The skewness of a r.v. measures the symmetry of a

dist. About its mean value.

{ }

continuous is X if

))((

discrete. is X if

)())((

)]([)(

3

3

3

1

3

3

3

x

x

x

n

i

ii

x

fXEX

XPXEX

XEXEXSkew

σ

σ

σ

∫

∑

∞

∞−

=

−

=

−=

−=


Kurtosis

� The kurtosis of a r.v. measures the thickness in the

tails of a distribution.

{ }

continuous is X if

))((

discrete. is X if

)())((

)]([)(

4

4

4

1

4

4

4

x

x

x

n

i

ii

x

fXEX

XPXEX

XEXEXKurt

σ

σ

σ

∫

∑

∞

∞−

=

−

=

−=

−=


Example

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.

0.250.50.25P(x)

210x


Covariance

� Covariance between two r.v.

{ }{ }[ ])()( YEYXEXEXY −−=σ


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.


Useful Formulas


� 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

Interpreting Covariance


Correlation Coefficient

� The Correlation Coefficient shows the strength of the linear association between two variables

where:

ρ = correlation coefficient (“rho”)

σxy = covariance between x and y

σx = standard deviation of variable x

σy = standard deviation of variable y

yx

yx

σσ

σρ =


� 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

Interpreting the Correlation Coefficient


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.


Independence

not true. is converse The

edness.uncorrelatE(X)E(Y)=E(XY)ceindependen Thus

eduncorrelat are Y and X then E(X)E(Y),=E(XY) If

E(X)E(Y)=E(XY)t then independen are Y and X If

R.y x,allfor

disc.r.v.for y)x)P(YP(Xy)andYx P(Xor

r.v.cont for (y)(x)ff=y)(x,f

ifonly and ift independen are Y and X r.v. The

YXYX,

⇒⇒

∈

=====


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


Linear Combinations of Two Random

Variables

� Let X~(µX,σX2 ) and Y~(µY,σY

2 ) 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 σX

2 +b2 σY2 +2abσXY=a2 σX

2 +b2 σY2

+2abσX σYρ


Linear Combinations of N Random

Variables


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 σX

2 +b2 σY2 +2abσX σYρ

10.50-0.5-1

3

2.5

2

1.5

1

0.5

0

Correlation

Sigma(Z)

Correlation

Sigma(Z)

Fec512.02

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