multivariate probability

7/29/2019 Multivariate Probability

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MULTIVARIATE PROBABILITY DISTRIBUTIONS

1. PRELIMINARIES

1.1. Example. Consider an experiment that consists of tossing a die and a coin at the sametime. We can consider a number of random variables defined on this sample space. Wewill assign an indicator random variable to the result of tossing the coin. If it comes upheads we will assign a value of one, and if it comes up zero we will assign a value of zero.Consider the following random variables.

X1: The number of dots appearing on the die.X2: The sum of the number of dots on the die and the indicator for the coin.X3: The value of the indicator for tossing the coin.X4: The product of the number of dots on the die and the indicator for the coin.

There are twelve sample points associated with this experiment where the first elementof the pair is the number on the die and the second is whether the coin comes up heads ortails.

E1 : 1H E2 : 2H E3 : 3H E4 : 4H E5 : 5H E6 : 6H

E7 : 1T E8 : 2T E9 : 3T E10 : 4T E11 : 5T E12 : 6T

Random variable X1 has six possible outcomes, each with probability16 . Random vari-

able X3 has two possible outcomes, each with probability12 . Consider the values of X2

for each of the sample points. The possible outcomes and the probabilities for X2 are asfollows:

TABLE 1. Probability ofX2

Value of Random Variable Probability

1 1/12

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

7 1/12

The possible outcomes and the probabilities for X4 are as follows:

Date: January 23, 2006.

1


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MULTIVARIATE PROBABILITY DISTRIBUTIONS 3

In the single-variable case, the probability function for a discrete random variable Xassigns non-zero probabilities to a countable number of distinct values ofX in such a waythat the sum of the probabilities is equal to 1. Similarly, in the bivariate case the jointprobability function p(x1, x2) assigns non-zero probabilities to only a countable numberof pairs of values (x1, x2). Further, the non-zero probabilities must sum to 1.

2.2. Properties of the Joint Probability (or Density) Function.

Theorem 1. IfX1 and X2 are discrete random variables with joint probability function p(x1, x2),then

(i) p(x1, x2) 0 for all x1, x2.(ii)

x1, x2

p(x1, x2) = 1 , where the sum is over all values (x1, x2) that are assigned non-zero

probabilities.

Once the joint probability function has been determined for discrete random variablesX1 and X2, calculating joint probabilities involving X1 and X2 is straightforward.

2.3. Example 1. Roll a red die and a green die. Let

X1 = number of dots on the red die

X2 = number of dots on the green die

There are 36 points in the sample space.

TABLE 4. Possible Outcomes of Rolling a Red Die and a Green Die. (Firstnumber in pair is number on red die.)

Green 1 2 3 4 5 6

Red1 1 1 1 2 1 3 1 4 1 5 1 62 2 1 2 2 2 3 2 4 2 5 2 63 3 1 3 2 3 3 3 4 3 5 3 64 4 1 4 2 4 3 4 4 4 5 4 65 5 1 5 2 5 3 5 4 5 5 5 66 6 1 6 2 6 3 6 4 6 5 6 6

The probability of (1, 1) is 136 . The probability of (6, 3) is also136 .

Now consider P(2 X1 3, 1 X2 2). This is given as

P(2 X1 3, 1 X2 2) = p(2, 1) + p(2, 2) + p(3, 1) + p(3, 2)

=4

36=

1

9


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4 MULTIVARIATE PROBABILITY DISTRIBUTIONS

2.4. Example 2. Consider the example of tossing a coin and rolling a die from section 1.Now consider P(2 X1 3, 1 X2 2). This is given as

P(2 X1 4, 3 X2 5) = p(2, 3) + p(2, 4) + p(2, 5)

+ p(3, 3) + p(3, 4) + p(3, 5)

+ p(4, 3) + p(4, 4) + p(4, 5)=

5

36

2.5. Example 3. Two caplets are selected at random from a bottle containing three aspirin,two sedative, and four cold caplets. If X and Y are, respectively, the numbers of aspirinand sedative caplets included among the two caplets drawn from the bottle, find the prob-abilities associated with all possible pairs of values ofX and Y?

The possible pairs are (0, 0), (0, 1), (1, 0), (1, 1), (0, 2), and (2, 0). To find the probabilityassociated with (1, 0), for example, observe that we are concerned with the event of gettingone of the three aspirin caplets, none of the two sedative caplets, and hence, one of the fourcold caplets. The number of ways in which this can be done is

31

20

41

= 12

and the total number of ways in which two of the nine caplets can be selected is9

2

= 36.

Since those possibilities are all equally likely by virtue of the assumption that the selec-tion is random, it follows that the probability associated with (1, 0) is 1236 =

13 . Similarly, the

probability associated with (1, 1) is

31

21

40

36

=6

36

=1

6and, continuing this way, we obtain the values shown in the following table:

TABLE 5. Joint Probability of Drawing Aspirin (X1) and Sedative Caplets (Y).

x

0 1 2

0 1613

112

y 1 2916 0

2

1

36 0 0

We can also represent this joint probability distribution as a formula


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p(x, y) =

3x

2y

42xy

36

, x = 0, 1, 2; y = 0, 1, 2; 0 (x + y) 2

3. DISTRIBUTION FUNCTIONS FOR DISCRETE MULTIVARIATE RANDOM VARIABLES

3.1. Definition of the Distribution Function. IfX1 and X2 are discrete random variables,the function given by

F(x1, x2) = P [X1 x1, X2 x2] =

u1x1

u2x2

p(u1, u2) < x1 <

< x2 < (2)

where p(u1, u2) is the value of the joint probability function of X1 and X2 at (u1, u2) iscalled the joint distribution function, or the joint cumulative distribution ofX1 and X2.

3.2. Examples.

3.2.1. Example 1. Consider the experiment of tossing a red and green die where X1 is thenumber of the red die and X2 is the number on the green die.

Now find F(2, 3) = P(X1 2, X2 3). This is given by summing as in the definition(equation 2).

F(2, 3) = P [X1 2, X2 3] =u12

u23

p(u1, u2)

= p(1, 1) + p(1, 2) + p(1, 3) + p(2, 1) + p(2, 2) + p(2, 3)

=1

36+

1

36+

1

36+

1

36+

1

36+

1

36

=6

36

=1

63.2.2. Example 2. Consider Example 3 from Section 2. The joint probability distribution isgiven in Table 5 which is repeated here for convenience.

TABLE 5. Joint Probability of Drawing Aspirin (X1) and Sedative Caplets (Y).

x

0 1 2

0 1613

112

y 1 2916 0

2 136 0 0

The joint probability distribution is


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4.5. Properties of the Joint Distribution Function.

Theorem 3. IfX1 and X2 are random variables with joint distribution function F(x1, x2), then

(i) F(, ) = F(, x2) = F(x1, ) = 0

(ii) F(, ) = 1(iii) Ifa < b and c < d, then F(a, c) < F(b, d)(iv) Ifa > x1 and b > x2, then F(a, b) F(a, x2) F(x1, b) + F (x1, x2) 0

Part (iv) follows because

F(a, b) F(a, x2) F(x1, b) + F(x1, x2) = P [x1 < X1 a, x2 < X2 b] 0

Note also that

F(

,) limx1 limx2

F(

x1

, x2) = 1

implies that the joint density function f(x1, x2) must be such that the integral off(x1, x2)over all values of(x1, x2) is 1.

4.6. Examples of a Joint Distribution Function and Density Functions.

4.6.1. Deriving a Distribution Function from a Joint Density Function. Consider a joint densityfunction for X1 and X2 given by

f(x1, x2) =

x1 + x2 0 < x1 < 1, 0 < x2 < 1

0 elsewhere

This has a positive value in the square bounded by the horizontal and vertical axesand the vertical and horizontal lines at one. It is zero elsewhere. We will therefore needto find the value of the distribution function for five different regions: second, third andfourth quadrants, square defined by the vertical and horizontal lines at one, area between

the vertical axis and a vertical line at one and above a horizontal line at one in the firstquadrant, area between the horizontal axis and a horizontal line at one and to the rightof a vertical line at one in the first quadrant, the area in the first quadrant not previouslymentioned. This can be diagrammed as follows.


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f2(x2) =

0

2x2ex1 dx1

= 2x2ex1

0

= 0

2x2e0

= 2x2e0

= 2x2

6.5. Example 2. Let the joint density of two random variables x and y be given by

f(x, y) =

16(x + 4y) 0 < x < 2, 0 < y < 1

0 otherwise

What are the marginal densities ofx and y?First find the marginal density for x.

fX(x) =

10

1

6(x + 4y) dy

=1

6(xy + 2y2)

10

=1

6(x + 2)

1

6(0)

=1

6(x + 2)

Now find the marginal density for y.

fY(y) =

2

0

16

(x + 4y) dx

=1

6

x2

2+ 4xy

20

=1

6

4

2+ 8y

1

6(0)

=1

6(2 + 8y)

7. CONDITIONAL DISTRIBUTIONS

7.1. Conditional Probability Functions for Discrete Distributions. We have previouslyshown that the conditional probability ofA given B can be obtained by dividing the prob-ability of the intersection by the probability ofB, specifically,


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TABLE 6. Joint and Marginal Probabilities ofX1 and X2.

X2

1 2 3 4 5 6 7

1 112112 0 0 0 0 0

16

2 0 112112 0 0 0 0

16

X1 3 0 0112

112 0 0 0

16

4 0 0 0 112112 0 0

16

5 0 0 0 0 112112 0

16

6 0 0 0 0 0 112112

16

112

16

16

16

16

16

112

TABLE 7. Probability Function for X1 given X2.

X2

1 2 3 4 5 6 7

1 1 12 0 0 0 0 016

2 0 1212 0 0 0 0

16

X1 3 0 0

1

2

1

2 0 0 0

1

6

4 0 0 0 1212 0 0

16

5 0 0 0 0 1212 0

16

6 0 0 0 0 0 12 116

112

16

16

16

16

16

112

7.3.1. Discussion. In the continuous case, the idea of a conditional distribution takes ona slightly different meaning than in the discrete case. IfX1 and X2 are both continuous,

P(X1 = x1 | X2 = x2) is not defined because the probability of any one point is identicallyzero. It make sense however to define a conditional distribution function, i.e.,

P(X1 x1 | X2 = x2)


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X2

1 2 3 4 5 6 7

1 1212 0 0 0 0 0

16

2 0 1212 0 0 0 0

16

X1 3 0 012

12 0 0 0

16

4 0 0 0 1212 0 0

16

5 0 0 0 0 1212 0

16

6 0 0 0 0 0 1212

16

112

16

16

16

16

16

112

because the value ofX2 is known when we compute the value the probability that X1 isless than some specific value.

7.3.2. Definition of a Continuous Distribution Function. IfX1 and X2 are jointly continuousrandom variables with joint density function f(x1, x2), then the conditional distributionfunction ofX1 given X2 = x2 is

F (x1 | x2) = P(X1 x1 | X2 = x2) (16)

We can obtain the unconditional distribution function by integrating the conditional oneover x2. This is done as follows.

F (x1) =

F(x1 | x2)fX2(x2) dx2 (17)

We can also find the probability that X1 is less than x1 is the usual fashion as

F (x1) =

x1

fX1(t1) dt1 (18)

But the marginal distribution inside the integral is obtained by integrating the joint den-sity over the range ofx2. Specifically,

fX1 (t1) =

fX1X2(t1, x2) dx2 (19)

This implies then that

F (x1) =

x1

fX1X2(t1, x2) dt1 dx2 (20)


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8.2.1. Discrete Random Variables. IfX and Y are discrete random variables with joint prob-ability density function p(x, y) and marginal density functions pX(x) and pY(y), respec-tively, then X and Y are independent if, and only if

pX,Y(x, y) = pX(x)pY(y)

= p(x)p(y) (27)

for all pairs of real numbers (x, y).

8.2.2. Continuous Bivariate Random Variables. IfX and Y are continuous random variableswith joint probability density function f(x, y) and marginal density functions fX(x) andfY(y), respectively then X and Y are independent if and only if

fX,Y(x, y) = fX(x)fY(y)

= f(x)f(y)(28)

for all pairs of real numbers (x, y).

8.3. Continuous Multivariate Random Variables. In a more general context the variablesX1, X2, . . . , X k are independent if, and only if

fX1, X2,...,Xk(x1, x2, . . . , xk) =k

i=1

fXi(xi)

= fX1(x1)fX2(x2) . . . f Xk(xk)

= f(x1)f(x2) . . . f (xk)

(29)

In other words two random variables are independent if the joint density is equal to theproduct of the marginal densities.

8.4. Examples.

8.4.1. Example 1 Rolling a Die and Tossing a Coin. Consider the previous example wherewe rolled a die and tossed a coin. X1 is the number on the die, X2 is the number ofthe die plus the value of the indicator on the coin (H = 1). Table 8 is repeated here forconvenience. For independence p(x, y) = p(x)p(y) for all values ofx1 and x2.

To show that the variables are not independent, we only need show that

p(x, y) = p(x)p(y)

Consider p(1, 2) =12 . If we multiply the marginal probabilities we obtain

1

6

1

6

=

1

36=

1

2


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X2

1 2 3 4 5 6 7

1 1212 0 0 0 0 0

16

2 0 1212 0 0 0 0

16

X1 3 0 012

12 0 0 0

16

4 0 0 0 1212 0 0

16

5 0 0 0 0 1212 0

16

6 0 0 0 0 0 1212

16

112

16

16

16

16

16

112

8.4.2. Example 2 A Continuous Multiplicative Joint Density. Let the joint density of tworandom variables x1 and x2 be given by

f(x1x2) =2x2e

x1 x1 0, 0 x2 1

0 otherwise

The marginal density for x1 is given by

f1(x1) =

10

2x2ex1 dx2

= x22ex11

0

= ex1 0

= ex1


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f2(x2) =

0

2x2ex1 dx1

= 2x2ex1

0

= 0

2x2e0

= 2x2e0

= 2x2

It is clear the joint density is the product of the marginal densities.

8.4.3. Example 3. Let the joint density of two random variables x and y be given by

f(x, y) =

3x2 log[y]

21 + log[2] 2log[4]0 x 1, 2 y 4

0 otherwise

First find the marginal density for x.

fX(x) =

10

3x2 log[y]

2

1 + log[2] 2log[4] dy

=3x2y

log[y] 1

2

1 + log[2] 2 log[4]

4

2

=3x24

log[4] 1

+ 3x22

log[2] 1

2

1 + log[2] 2log[4]

=

3x2

2

log[2] 1

4

log[4] 1

2

1 + log[2] 2 log[4]

=3x2

2 log[2] 2 4 log[4] + 4

2

1 + log[2] 2 log[4]

=3x2

2 log[2] 4 log[4] + 2

2

1 + log[2] 2 log[4]

=3x2

2

1 + log[2] 2 log[4]

2

1 + log[2] 2log[4]

= 3x2

Now find the marginal density for y.


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We can show that this is a proper density function by integrating it over the range of xand y.

0

x

2e(x+y) dy dx =

0

2e(x+y)

x

dx

=

0

2e2x dx

= e2x0

= e

e0

= 0 + 1 = 1

Or in the other order as follows.

0

y

0

2e(x+y) dxdy =

02e(x+y)y

0 dy

=

0

2ey 2e2y

dy

= 2ey0

ey

0

=

2e + 2

e + 1

= [0 + 2] [0 + 1]

= 2 1 = 1


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9.1.1. Discrete Case. Let X = (X1, X2, . . . , X k) be a k-dimensional discrete random vari-able with probability function p(x1, x2, . . . , xk). Let g(, , . . . , ) be a function of the krandom variables (X1, X2, . . . , X k). Then the expected value ofg(X1, X2, . . . , X k) is

Eg(X1, X2, . . . , X k) =xkxk1

x2x1

g(x1, . . . , xk)p(x1, x2, . . . , xk) (30)

9.1.2. Continuous Case. Let X = (X1, X2, . . . , X k) be a k-dimensional random variablewith density f(x1, x2, . . . , xk). Let g(, , . . . , ) be a function of the k random variables(X1, X2, . . . , X k). Then the expected value ofg(X1, X2, . . . , X k) is

E[g(X1, X2 . . . , X k)] =

xk

xk1

. . .

x2

x1

g(x1, . . . , xk)fX1, . . . ,Xk(x1, . . . , xk)dx1 . . . dxk

=

. . .

g(x1, . . . , xk)fX1, . . . ,X k(x1, . . . , xk) dx1 . . . d xk

(31)

if the integral is defined.Similarly, ifg(X) is a bounded real function on the interval [a, b] then

E(g(X)) =

ba

g(x) dF(x) =

ba

g dF (32)

where the integral is in the sense of Lebesque and can be loosely interpreted as f(x) dx.Consider as an example g(x1, . . . , xk) = xi. Then

E[g(X1, . . . , X k)] E[Xi]

. . .

xif(x1, . . . , xk) dx1 . . . d xk

xifXi(xi) dxi

(33)

because integration over all the other variables gives the marginal density ofxi.

9.2. Example. Let the joint density of two random variables x1 and x2 be given by

f(x1x2) =2x

2ex1 x

1 0, 0 x

2 1

0 otherwise



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E[X1] =

10

2x2 dx2

= x221

0= 1

Now find it using the marginal density of x1. Integrate as follows:

E[X1] =

0

x1ex1 dx1

We will need to use a u dv substitution to evaluate the integral. Let

u = x1 and dv = ex1 dx1

then

du = dx1 and v = e

x1

Then

0

x1ex1 dx1 = x1e

x10

0

ex1 dx1

= x1ex10

+

0

ex1 dx1

= 0 + ex10

= e e0

= 0 + 1 = 1

We can likewise show that the expected value of x2 is23 . Now consider E[x1x2]. We can

obtain it as

E[X1X2] =

10

0

2x1x22ex1 dx1 dx2

Consider the inside integral first. We will need a u dv substitution to evaluate the inte-gral. Let

u = 2x1x22 and dv = e

x1 dx1

then

du = 2x22 dx1 and v = ex1


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Then 0

2x1x22ex1 dx1 = 2x1x

22ex10

0

2x22ex1 dx1

= 2x1x22ex10 +

0

2x22ex1 dx1

= 0 + 2x22ex10

= 2x22

Now integrate with respect to x2.

E[X1X2] =

10

2x22 dx2

=2

3

x321

0

=2

3

9.3. Properties of Expectation.

9.3.1. Constants.

Theorem 5. Let c be a constant. Then

E[c]

x

y

cf(x, y) dy dx

cx

y f(x, y) dydx

c

(34)

9.3.2. Theorem.

Theorem 6. Let g(X1, X2) be a function of the random variables X1 and X2 and let a be aconstant. Then

E[ag(X1, X2)]

x1

x2

ag(x1, x2)f(x1, x2) dx2 dx1

ax1

x2

g(x1, x2)f x1, x2) dx2 dx1

aE[g(X1, X2)]

(35)


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E

g(X1)h(X2)

x1

x2

g(x1)h(x2)f(x1, x2) dx2 dx1

x1x2 g(x1)h(x2)fX1(x1)fX2(x2) dx2 dx1

x1

g(x1)fX1(x1)

x2

h(x2)fX2(x2) dx2

dx1

x1

g(x1)fX1(x1)

E[h(X2)]

dx1

E[h(X2)]

x1

g(x1)fX1(x1) dx1

E[h(X2)] E[g(X1)]

(40)

10. VARIANCE, COVARIANCE AND CORRELATION

10.1. Variance of a Single Random Variable. The variance of a random variable X withmean is given by

var(X) 2 E

X E(X)2

E

(X )2

(x )2f(x) dx

x2f(x) dx

xf(x)dx

2

E(x2) E2(x)

(41)

The variance is a measure of the dispersion of the random variable about the mean.

10.2. Covariance.

10.2.1. Definition. Let X and Y be any two random variables defined in the same proba-bility space. The covariance of X and Y, denoted cov[X, Y] or X, Y, is defined as


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Then compute the expected value ofX2 as follows.

E[X2] =

20

2x2

3

8x1x2 dx1 dx2

= 20 3

16x2

1x22x2 dx

2

=

20

12

16x2

3

16x32

dx2

=

20

3

4x2

3

16x32

dx2

=

3

8x22

3

64x42

2

0

=12

8

48

64

=96

64

48

64=

48

64

=3

4

The covariance is then given by

cov[X, Y] E[XY] E[X]E[Y]

6

5

3

2

3

4

6

5

9

8

48

40

45

40

=3

40

10.3. Correlation. The correlation coefficient, denoted by [X, Y], or X,Y of random vari-ables X and Y is defined to be

X,Y =cov[X, Y]

XY(43)

provided that cov[X, Y], X and Y exist, and X, Y are positive. The correlation coeffi-cient between two random variables is a measure of the interaction between them. It alsohas the property of being independent of the units of measurement and being boundedbetween negative one and one. The sign of the correlation coefficient is the same as the


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These random variables are not independent because the joint probabilities are not theproduct of the marginal probabilities. For example

pX1X2 [1, 1] =1

16= pX1(1)pX1(1) =

5

16

5

16

=

25

256

Now compute the covariance between X1 and X2. First find E[X1] as follows

E[X1] = (1)

5

16

+ (0)

6

16

+ (1)

5

16

= 0

Similarly for the expected value ofX2.

E[X2] = (1)

5

16

+ (0)

6

16

+ (1)

5

16

= 0

Now compute E[X1X2] as follows

E[X1X2] = (1)(1) 1

16

+ (1)(0) 3

16

+ (1)(1) 1

16

+ (0)(1)

3

16

+ (0)(0)(0) + (0)(1)

3

16

+ (1)(1)

1

16

+ (1)(0)

3

16

+ (1)(1)

1

16

=1

16

1

16

1

16+

1

16= 0

The covariance is then

cov[X, Y] E[XY] E[X]E[Y]

0 (0) (0) = 0

In this case the covariance is zero, but the variables are not independent.

10.5. Sum of Variances var[a1x1 + a2x2].

var[a1x1 + a2x2] = a21 var(x1) + a

22 var(x2) + 2a1a2 cov(x1, x2)

= a2121 + 2a1a212 + a

22

22

= [a1 a2]

21 12

21 22

a1

a2

= var

[a1 a2]

x1x2

(49)


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We can find the marginal density ofy by integrating the joint density with respect to xas follows

fY(y) =

f(x, y) dx

=

1y

02 dx

= 2x1y0

= 2(1 y), 0 y 1

We find the conditional density ofX given that Y = y by forming the ratio

fX|Y(x | y) =f(x, y)

fY(y)

=2

2(1 y)

=1

(1 y), 0 x 1 y

We then form the expected value by multiplying the density by x and then integratingover x.

E[X | Y] =1y0 x

1

(1 y) dx

=1

(1 y)

1y0

x dx

=1

(1 y)

x2

2

1y

0

=1

(1 y)

(1 y)2

2

=(1 y)

2

We can find the unconditional expected value ofX by multiplying the marginal densityofy by this expected value and integrating over y as follows


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We find the expected value of X by multiplying the conditional density by x and thenintegrating over x

E[X] =

10

1

4x (4x + 2) dx

=10

14

4x2 + 2x

dx

=1

4

4

3x3 + x2

1

0

=1

4

4

3+ 1

=

1

4

7

3

=7

12

(71)

To find the variance ofX, we first need to find the E[X2]. We do this as follows

E[X2] =10

14

x2 (4x + 2) dx

=

10

1

4

4x3 + 2x2

dx

=1

4

x4 +

2

3x3

1

0

=1

4

1 +

2

3

=

1

4

5

3

=

5

12

(72)

The variance ofX is then given by

var(X) E

(X E(X))2

E(x2) E2(x)

=5

12

7

12

2

=5

12

49

144

= 60144 49144

=11

144

(73)


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E[X] = EY

E[X | Y]

=

20

(4 + 3y)

(6 + 6y)

1

4(1 + y) dy

=1

4

2

0

(4 + 3y)(1 + y)

6(1 + y)dy

=1

24

20

(4 + 3y) dy

=1

24

4y +

3

2y2

2

0

=1

24(8 + 6)

=

14

24 =

7

12

(76)

We find the conditional variance by finding the expected value of X2 given Y and thensubtracting the square ofE[X | Y].

E[X2 | Y] =

10

x2(2x + y)

(1 + y)dx

=1

1 + y

10

(2x3 + x2y) dx

=1

1 + y1

2

x4 +1

3

x3y1

0

=1

1 + y

1

2+

1

3y

=

12 +

y3

1 + y

=

1

6

3 + 2y

1 + y

(77)

Now square E[X | Y].

E

2

[X | Y] =1

6(4 + 3y)

(1 + y)2

=1

36

(4 + 3y)2

(1 + y)2

(78)


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Now substitute in equation 90 into equation 88 as follows

EY

E[X | Y]

2

=

1

144

20

9y2 + 24y + 16

1 + ydy

=1

144

20

9y + 15 +1

1 + ydy

=1

144

9y22

+ 15y + log[y + 1]

2

0

=1

144

362

+ 30 + log[3]

=1

144

48 + log[3]

(91)

The variance is obtained by subtracting the square of (87) from (91)

var

E[X | Y]

= EY

E[X | Y]

2+

EY

E[X | Y]2

=1

144

48 + log[3]

7

12

2

=1

144

48 + log[3]

49

144

=1

144log[3] 1

(92)

We can show that the sum of (85) and (92) is equal to the var[X1] as in Theorem 13:

var[X] = E

var[X | Y = y]

+ var

E[X | Y = y]

=1

144

log[3] 1

+

1

144

12 log[3]

=

log[3] 1 + 12 log[3]

144

= 11144

(93)

which is the same as in equation 73.


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12. CAUCHY-S CHWARZ INEQUALITY

12.1. Statement of Inequality. For any functions g(x) and h(x) and cumulative distribu-tion function F(x),the following holds:

g(x)h(x) dF(x)

g(x)

2

dF(x)

1

2 h(x)

2

dF(x)

1

2

(94)

where x is a vector random variable.

12.2. Proof. Form a linear combination of g(x) and h(x), square it and then integrate asfollows:

tg(x) + h(x)2

dF(x) 0 (95)

The inequality holds because of the square and dF(x) > 0. Now expand the integrandin (95) to obtain

t2 g(

x)2 dF

(x

) + 2t g

(x

)h

(x

)dF

(x

) +h

(x

)2 dF

(x

) 0 (96)This is a quadratic equation in t which holds for all t. Now define t as follows:

t =

g(x)h(x) dF(x)g(x)

2dF(x)

(97)

and substitute in (96)

g(x)h(x) dF x)

2g(x)

2dF(x)

2

g(x)h(x) dF(x)

2g(x)

2dF(x)

+

h(x)

2dF(x) 0

g(x)h(x) dF(x)2

g(x)2

dF(x)

h(x)

2 dF(x)

g(x)h(x) dF(x)

2

h(x)

2dF(x)

g(x)

2dF(x)

g(x)h(x) dF(x)

h(x)2

dF(x) 1

2

gx)2

dF(x)2

(98)

12.3. Corollary 1. Consider two random variables X1 and X2 and the expectation of theirproduct. Using (98) we obtain

E(X1X2)

2 E(X21 )E(X

22 )E(X1X2) E(X21 ) 12 E(X22 ) 12

(99)


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12.4. Corollary 2.

| cov(X1X2)|

multivariate probability

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