Chapter 2Multivariate Distributions
Math 6203Fall 2009
Instructor: Ayona Chatterjee
Random Vector
• Given a random experiment with a sample space C. Consider two random variables X1 and X2 which assign to each element c of C one and only one ordered pair of numbers X1(c)=x1 and X2(c)=x2. Then we say that (X1, X2) is a random vector.
• The space of (X1, X2) is the set of ordered pairs D={(x1, x2) : X1(c)=x1 and X2(c)=x2 }
Cumulative Distribution Function
• The joint cumulative distribution function of (X1, X2) is denoted by FX1,X2 (x1, x2) and is given as FX1,X2 (x1, x2) =P[X1≤x1, X2 ≤x2)].
• A random vector (X1, X2 )is a discrete random variable is its space D is finite or countable.
• A random vector (X1, X2 ) with space D is continuous if its cdf FX1,X2 (x1, x2) is continuous.
Probability Mass Function
• For discrete random variables X1 and X2, the joint pmf is defined as
DXX
XX
XX
xxp
xxpNote
xXxXPxxp
1),(*
1),(0*
],[),(
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21
Probability Density Function
• For a continuous random vector
DXX
XX
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x x
XXXX
dxdxxxf
xxfNote
xxfxxxxF
dwdwwwfxxF
1),(*
0),(*
),(),(
),(),(
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212
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1 2
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Marginals
• The marginal distributions can be obtained from the joint probability density function.
• For a discrete and continuous random vector the marginals can be obtained as below:
2211
211
),()(
),()(
211
2
211
dxxxfxf
xxpxp
XXX
xXXX
Expectation
• Suppose (X1, X2) is of the continuous type. Then E(Y) exists if
212121
212121
),(),()(
),(),(
21
21
dxdxxxfxxgYE
then
dxdxxxfxxg
XX
XX
Theorem
• Let (X1, X2) be a random vector. Let Y1 = g1(X1, X2) and Y2 = g2 (X1, X2) be a random variable whose expectations exits. Then for any real numbers k1 and k2.
E(k1 Y1 + k2 Y2 )= k1E(Y1 ) + k2 E(Y2 )
Note
222212122 )()(),()())((221
dxxfxgdxdxxxfxgXgE XXX
Moment Generating Function
• Let X = (X1. X2 )’ be a random vector. If E(et1x1+t2x2 ) exists for |t1 |<h1 and |t2 |<h2 where h1 and h2 are positive, the mgf is given as
.X of mgf theis ),0( and X of mgf theis )0,(),(
][)(
2211
'21
2121
'
21
tMtMttt
where
eEtM
XXXX
XtXX
2.3 CONDITIONAL DISTRIBUTIONS AND EXPECTATIONS
• So far we know– How to find marginals given the joint distribution.
• Now– Look at conditional distribution, distribution of
one of the random variable when the other has a specific value.
Conditional pmf
• We define
• SX2 is the support of X2.
• Here we assume pX1 (x1) > 0.• Thus conditional probability is the joint
divvied by the marginal.
2
1
2,1
12
2
1
21
11
221112| )(
),(
)(),()|(
X
X
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Sx
xp
xxp
xXPxXxXPxxp
Conditional pdf
• Let fX1x2 (x1, x2 ) be the joint pdf and fx1 (x1) and fx2 (x2) be the marginals for X1 and X2 respectively then the conditional pdf of X2, given X1 is
0)(
)|()(
),()|(
1
121|21
2112|
1
1
2,1
12
xf
xxfxf
xxfxxf
X
X
XXXX
Note
1)|(*
0)|(*
2121|2
121|2
dxxxf
xxf
Conditional Expectation and Variance
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)]|([)|()|var(
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bygiven isexists,it ifX given that , ofn expectatio lconditiona the,X offunction a is If
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Theorem
• Let (X1, X2) be a random vector such that the variance of X2 is finite. Then – E[E(X2 |X1)]=E(X2)
– Var[E(X2 |X1 )]≤ var(X2 )
2.4 The Correlation Coefficient
21
21
21
),()])([(),(
)()()()])([(
YXCovYXEYXCov
YEXEXYEYXE
Here ρ is called the correlation coefficient of X and Y. Cov(X,Y) is the covariance between X and Y.
The Correlation Coefficient
• Note that -1 ≤ ρ≤ 1.• For the bivariate case– If ρ = 1, the graph of the line y = a + bx (b > 0) contains
all the probability of the distribution of X and Y.– For ρ = -1, the above is true for the line y = a + bx with
b < 0.– For the non-extreme case, ρ can be looked as a
measure of the intensity of the concentration of the probability of X and Y about a line y = a + bx.
Theorem
• Suppose (X,Y) have a joint distribution with the variance of X and Y finite and positive. Denote the means and variances of X and Y by µ1 , µ2 and σ1
2 , σ22 respectively, and let ρ be the
correlation coefficient between X and Y. If E(Y|X) is linear in X then
)1())|(var(
)()|(
222
11
22
XYE
XXYE
2.5 Independent random Variables
• If the conditional pdf f2|1 (x2|x1) does not depend upon x1 then the marginal pdf of X2 equals the conditional pdf f2|1 (x2|x1) .
• Let the random variables X and Y have joint pdf f(x,y) and the marginals fx (x) and fy (y) respectively. The random variables X and Y are said to be independent if and only if – f(x,y)= fx (x) fy (y) – Similar defintion can be wriiten for discrete random variables.– Random variables that are not independent are said to be
dependent.
Theorem
• Let the random variables X and Y have support S1 and S2, respectively and have the joint pdf f(x,y). Then X and Y are independent if and only if f(x,y) can be written as a product of a nonnegative function of x and a nonnegative function of y. That is f(x,y)=g(x)h(y) where g(x)>0 and h(y)>0.
Note
• In general X and Y must be dependent of the space of positive probability density of X and Y is bounded by a curve that is neither a horizontal or vertical line.
• Example; f(x,y)=8xy, 0< x< y < 1– S={(x,y): 0< x< y < 1} This is not a product space.
Theorems
• Let (X, Y) have the joint cfd F(x,y) and let A and Y have the marginal cdfs Fx (x) and Fy (y) respectively. Then X and Y are independent if and only if – F(x,y)= Fx (x)Fy (y)
• The random variable X and Y are independent if and only if the following condition holds.– P(a < X≤ b, c < Y ≤ d)= P(a < X≤ b)P( c < Y ≤ d)– For ever a < b, c < d and a,b,c and are constants.
Theorems
• Suppose X and Y are independent and that E(u(X)) and E(v(Y)) exist, then – E[u(x), v(Y)]=E[u(X)]E[v(Y)]
• Suppose the joint mgf M(t1,t2) exists for the random variables X and Y. Then X and Y are independent if and only if– M(t1,t2) = M(t1,0)M(0,t2) • That is the joint mfg if the product of the marginal
mgfs.
Note
• If X and Y are independent then the correlation coefficient is zero.
• However a zero correlation coefficient does not imply independence.
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