Joint Probability Distributions 2

8/3/2019 Joint Probability Distributions 2

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Joint Probability Distributions-

Part II


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Joint Distributions

Joint cumulative probability function:

The joint cdf of X,Y at a point (x,y) is just the

probability that both Xex and Y ey.

yYxXPyxF YX ee!,,


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Joint Distributions

Joint cumulative probability function:

If X,Y are independent, then the joint cdf of X

and Y is equal to the product of the cdf of Xand the cdf of Y

yFxF

yYPxXP

yYxXPyxF

YX

YX

!

ee!

ee!,,


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Joint Distributions

Covariance

If two variables have covariance 0, they may

or may not be independent

X

Y X

0 1 2

Y 0 0.00 0.17 0.00

1 0.33 0.00 0.33

2 0.00 0.17 0.00

0

0 1 2

1

2


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Joint Distributions

Sums and differences

The expectation of sums is the sum of

expectations

YcEXbEacYbXaE

YEXEYXE

!

!


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Joint Distributions


The variance of a sum is the sum of

variances, plus twice the covariance

YXbcCovYVcXVbcYbXaV

YXCovYVXVYXV

,2

,2

22!

!


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Joint Distributions


If two variables are uncorrelated (covariance

is 0), the variance of a sum is the sum of

variances

YVcXVbcYbXaV

YVXVYXV

22!

!


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Joint Distributions

Generalization

- The expectation of a sum is sum of the

expectations : E(X+Y+Z) = E(X) +E(Y) + E(Z) The variance of a sum is the sum of

variances plus twice the sum of

every possible covariance:

ZYCovZXCovYXCov

ZVYVXVZYXV

,2,2,2

!


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Joint Distributions

Generalization

- The expectation of a sum is sum of the

expectations :E(a+bX+cY+dZ) = a +b E(X) +c E(Y) +d E(Z)

- The variance of a sum is the sum of

variances plus twice the sum of

every possible covariance:

ZYcdCovZXbdCovYXbcCov

ZVdYVcXVbdZcYbXaV

,2,2,2

222

!


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Joint Distributions

Example: portfolio theory

Suppose there are two stocks AAA,BBB:

Stock E(R) Var(R) Cov

AAA 10 25 1

BBB 10 25


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Joint Distributions


A portfolio allocated 100% to stock AAA would

return 10% with a SD of 5%

A portfolio allocated 100% to stock BBB would

return 10% with SD 5%


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Joint Distributions


A portfolio allocated 50% to stock AAA and

50% to stock BBB:

10.010.05.010.05.0 !!

!BBBBBBAAAAAA

RERERE [[


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Joint Distributions


A portfolio allocated 50% to stock AAA and

50% to stock BBB:

The SD of return for this portfolio is 3.6% eventhough its expected return is 10%.

So, its a better portfolio its less risky!!!

13.001.05.05.0225.05.025.05.0

,2

22

22

!!

!BBBAAABBBAAABBBBBBAAAAAA

RRCovRVRVRV [[[[


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Assignment

Choose three stocks and collect data regarding theirclosing prices Pt for a period of 101 days starting from1/1/2009

Compute the daily return

for each of these stocks Compute the expected returns, variances and

covariances of these stocks.

Determine an optimal portfolio in the sense that it will

provide maximum return subject to the risk being lessthan a certain fixed quantity () (use Excel-Solver orsome such software).

Vary and study the changes in portfolio allocation.

1001

v

!

t

tt

t

P

PPR


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Convolution Formula for Non-negative

Integer-valued Random Variables

Let X and Y be two independent non-negative

integer valued random variables. Then

!

!!!!k

i

ikYPiXPkYXP

0

)()()(


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Bernoulli Distribution

If X1 and X2 are independent Ber(p) r.v.s

then X1+ X2 ~ Bin(2,p)


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Reproductive Property of Binomial

Distribution

If X1~ Bin(n1,p) and X2 ~ Bin(n2,p) and they

are independent then X1+ X2 ~ Bin(n1+n2,p)


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Reproductive Property of Poisson

Distribution

If X1~ Poi(1) and X2 ~ Poi(2) and they are

independent then X1+ X2 ~ Poi(1+2)


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Geometric Distribution

If X1 and X2 are independent Geo(p) r.v.s

then X1+ X2 ~ NB(2,p)


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Reproductive Property of Negative

Binomial Distribution

If X1~ NB(r1,p) and X2 ~ NB(r2,p) and they

are independent then X1+ X2 ~ NB(r1+r2,p)


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Understanding Correlation from

Scatter Plots

Examples of some real life data sets


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Example

(1) Change in Gross Private Investment is Correlated with

Change in GNP

-0.02

0.00

0.02

0.04

0.06

-0.2 -0.1 0.0 0.1 0.2

PCGNP

PCGPI


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Example

(2) Yield on Three Month Commercial Paper is Highly

Correlated with the Yield on One Year Treasury Bond

0

5

10

15

20

0 5 10 15 20

CPAP3M

TREASURY1Y


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Example

(3) Long Term Government Bond (LTGB) and Short Term

Treasury Returns (TBILL) Exhibited Little Correlation

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.000 0.005 0.010 0.015

LTGB

TBILL


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Example

(4) Long Term Corporate Bonds Returns (LTCB) and Stock

Markets Returns were Slightly Positively Correlated

-0.10

-0.05

0.00

0.05

0.10

0.15

-0.4 -0.2 0.0 0.2 0.4 0.6

LTCB

SP500


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Example

(5) Graduate Rankings ofUS Business Schools were

uncorrelated with Recruiter Rankings in 1988

0

5

10

15

20

25

0 5 10 15 20 25

RECRUITER

GRADUATE


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Example

(6) Coffee Price and Coffee Consumption are Negatively

Correlated

50

100

150

200

250

8 10 12 14 16 18 20

PRICE

CONS

Joint Probability Distributions 2

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