stat 712 fa 2021 lec 10 slides transformations of multiple

STAT 712 fa 2021 Lec 10 slides

Transformations of multiple random variables

Karl B. Gregory

University of South Carolina

These slides are an instructional aid; their sole purpose is to display, during the lecture,definitions, plots, results, etc. which take too much time to write by hand on the blackboard.

They are not intended to explain or expound on any material.

Karl B. Gregory (U. of South Carolina) STAT 712 fa 2021 Lec 10 slides 1 / 15

X X Y X Ä Y Y

Bivariate transformations

1 Bivariate transformations

2 Sums of independent random variables



We often wish to find the distribution of a function of two (or more) rvs:

Exercise: Let X1 and X2 be indep. Exponential(�) rvs. Let Y = X1/(X1 + X2).1 Find the cdf of Y .2 Find the pdf of Y .


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Theorem (Bivariate transformation method)

Let (X1,X2) be a pair of cont. rvs with joint pdf fX1,X2 on X and

Y1 = g1(X1,X2) and Y2 = g2(X1,X2),

where g1 and g2 define a 1:1 transformation of X onto Y (define these).

Let g�11 and g�1

2 be the functions satisfying

y1 = g1(x1, x2)y2 = g2(x1, x2)

() x1 = g�11 (y1, y2)

x2 = g�12 (y1, y2)

for all (x1, x2) 2 X and (y1, y2) 2 Y.

Then the joint pdf of (Y1,Y2) is given by

fY1,Y2(y1, y2) = fX1,X2(g�11 (y1, y2), g

�12 (y1, y2))|J(y1, y2)|, for (y1, y2) 2 Y,

where J(y1, y2) is the Jacobian (next slide), if J(y1, y2) is not always 0 on Y.


Jacobian


JacobianIn the setup of the previous slide, the Jacobian of the transformation is defined as

J(y1, y2) =

��

@@y1

g�11 (y1, y2)

@@y2

g�11 (y1, y2)

@@y1

g�12 (y1, y2)

@@y2

g�12 (y1, y2)

�� .

For real numbers a, b, c , d ,��a bc d

�� = ad � bc .

This is called the determinant.



Exercise: Let X1,X2 be independent Normal(0, 1) rvs.1 Find the joint pdf of Y1 = X1/X2 and Y2 = X2.2 Find the marginal pdf of Y1.


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Exercise: Let X1,X2 have joint pdf

fX1,X2(x1, x2) =1�2 exp

�x1 + x2

�

�1(x1 > 0, x2 > 0).

1 Find the joint pdf of Y1 = X1/(X1 + X2) and Y2 = X1 + X2.2 Find the marginal pdf of Y1.



Exercise: Let X1 ⇠ Beta(1, 1) and X2 ⇠ Beta(2, 1) be independent rvs.1 Find the joint pdf of Y1 = X1X2 and Y2 = X2.2 Find the marginal pdf of Y1.


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Exercise: Let Z1,Z2 have the bivariate Normal distribution, with joint pdf

fZ1,Z2(z1, z2) =12⇡

1p1 � ⇢2

exp

�1

2z21 � 2⇢z1z2 + z2

21 � ⇢2

�.

1 Find the joint pdf of U1 = Z1 + Z2 and U2 = Z1 � Z2.2 Find the marginal pdfs of U1 and U2.


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Exercise: Let Z1,Z2 have the bivariate Normal distribution, with joint pdf

fZ1,Z2(z1, z2) =12⇡

1p1 � ⇢2

exp

�1

2z21 � 2⇢z1z2 + z2

21 � ⇢2

�.

1 Find the joint pdf of U1 = min{Z1,Z2} and U2 = max{Z1,Z2}? (Not 1:1).2 Find the marginal pdf of U2 (work on this at home).


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Sums of independent random variables




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We now extend our definition of indepence to a collection of more than two rvs.

Mutual independence for a collection of rvsLet the rvs X1, . . . ,Xn have joint pdf/pmf f (x1, . . . , xn) and marginal pdfs/pmfssuch that Xi ⇠ fXi , i = 1, . . . , n. If

f (x1, . . . , xn) =nY

i=1

fXi (xi ) for all (x1, . . . , xn) 2 Rn,

we say that X1, . . . ,Xn are mutually independent rvs.

Expectation of a product of functionsIf X1, . . . ,Xn are mutually independent, then

E(g1(X1) · · · · · gn(Xn)) =Qn

i=1 Egi (Xi )

for any functions g1, . . . , gn : R ! R.


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Theorem (mgf method for sums of independent rvs)Let X1, . . . ,Xn be ind. rvs with mgfs MX1 , . . . ,MXn , resp. Let Y = X1 + · · ·+ Xn.

The mgf of Y is given by

MY (t) =nY

i=1

MXi (t).

Moreover, if X1, . . . ,Xn are ind. and all have mgf MX (are iid), then

MY (t) = [MX (t)]n.

Exercise: Prove the above.


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Exercise: Let X1, . . . ,Xn be ind. chi-squared rvs with dfs ⌫1, . . . , ⌫n, resp. Findthe distribution of Y = X1 + · · ·+ Xn.


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Exercise: Let X1, . . . ,Xn be ind. Normal rvs with means µ1, . . . , µn and variances�2

1 , . . . ,�2n , resp.

1 Find the distribution of Y = X1 + · · ·+ Xn.2 Find the distribution of V = a1X1 + · · ·+ anXn, for a1, . . . , an 2 R.3 Find the distribution of X̄n = (X1 + · · ·+ Xn)/n.


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Multivariate transformations, including non-1:1


2 Multivariate transformations, including non-1:1




We first define the joint pdf/pmf of a collection of d random variables.

Multivariate pdf/pmfThe joint pdf of a d-tuple of continuous rvs (X1, . . . ,Xd) is the functionf : Rd ! [0,1) such that

P((X1, . . . ,Xd) 2 A) =

Z· · ·

Z

Af (x1, . . . , xd)dx1 · · · dxd

for all A 2 B(R).

If X1, . . . ,Xd are discrete, then their joint pmf is the function p given by

p(x1, . . . , xd) = P(X1 = x1, . . . ,Xd = xd) for all (x1, . . . , xd) 2 Rd .


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We extend our definition of independence to a collection of more than two rvs.

Mutual independence for a collection of rvsLet the rvs X1, . . . ,Xd have joint pdf/pmf f (x1, . . . , xd) and marginal pdfs/pmfssuch that Xi ⇠ fXi , i = 1, . . . , d . If

f (x1, . . . , xd) =Qd

j=1 fXj (xj) for all (x1, . . . , xd) 2 Rd ,

we say that X1, . . . ,Xd are mutually independent rvs.

Theorem (Quick check for mutual independence)The random variables X1, . . . ,Xd with joint pdf/pmf f are mutually independentif and only if there exist functions g1, . . . , gd such that

f (x1, . . . , xd) =Qd

j=1 gj(xj) for all (x1, . . . , xd) 2 Rd .


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Multivariate transformationLet (X1, . . . ,Xd) have joint pdf f with support A. Consider the rvs

U1 = g1(X1, . . . ,Xd) . . . Ud = gd(X1, . . . ,Xd).

for some functions g1, . . . , gd : Rd ! Rd such that, given a partitionA0,A1, . . . ,Am of A, where P((X1, . . . ,Xd) 2 A0) = 0, the transformation is1 : 1 on each of the sets A1, . . . ,Am.

Let the inverse transformations be given by g�11k , . . . , g�1

dk , k = 1, . . . ,m such that

x1 = g�11k (u1, . . . , ud) . . . xd = g�1

1k (u1, . . . , ud)

is the unique value in Ak mapped to (u1, . . . , ud). Then

fU1,...,Ud (u1, . . . , ud) =mX

k=1

f (g�11k (u1, . . . , ud), . . . , g

�1dk (u1, . . . , ud))|Jk(u1, . . . , ud)|,

where Jk(u1, . . . , ud) is the Jacobian of the transformation on Ak , k = 1, . . . ,m.


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Exercise: Let X ⇠ Normal(0, 1). Find the pdf of U = X 2.


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Exercise: Let Z1,Z2ind⇠ Normal(0, 1) and consider the rvs

Y1 = Z 21 + Z 2

2

Y2 = Z1/q

Z 21 + Z 2

2 .

1 Find the joint pdf of (Y1,Y2).2 Check whether Y1 and Y2 are independent.3 Find the marginal pdf of Y2.


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Exercise: Let X1,X2,X3ind⇠ Exponential(1) and consider

Y1 =X1

X1 + X2

Y2 =X1 + X2

X1 + X2 + X3

Y3 = X1 + X2 + X3.

1 Find the joint pdf of (Y1,Y2,Y3).2 Check whether Y1, Y2, and Y3 are mutually independent.3 Find the marginal distributions Y1, Y2, and Y3.


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stat 712 fa 2021 lec 10 slides transformations of multiple

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