The binormal joint density function is
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![Problem 4.27: The binormal joint density function is
1
1
fx,y (x, y)
exp
2(1 – p2)
2n0102/1- p2
[(r – m1)²
of
(y – m2)? 2p(x – m1)(y – m2)]|
exp{-(z – m)C-'(z – m)'}
(27)"det(C)
where z = [r y], m = [m1 m2] and
of
C =
c-( )
ρσισ
1) With
c-()
4.
C =
9
we obtain of = 4, o = 9 and pơ102 = -4. Thus
2) The transformation Z = 2X +Y, W = X – 2Y is written in matrix notation as
()-(: )G)-()
2
1
= A
W
The ditribution fz,w(z, w) is binormal with mean m' = mA', and covariance matrix C' =
ACA'. Hence
C' = ({
9 2
2
4 -4
2
1
1 -2
9
-2
2 56
The off-diagonal elements of C' are equal to po zow = COV(Z,W). Thus COV (Z, W) = 2.
3) Z will be Gaussian with variance o? = 9 and mean
%3D
mz = [ m1
m2
= 4
21](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa80eb220-b198-4f3d-81bb-2cbd72fe9990%2Ffba5fc90-9255-42b5-ad8b-53172751698d%2F78yam6_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 4.27: The binormal joint density function is
1
1
fx,y (x, y)
exp
2(1 – p2)
2n0102/1- p2
[(r – m1)²
of
(y – m2)? 2p(x – m1)(y – m2)]|
exp{-(z – m)C-'(z – m)'}
(27)"det(C)
where z = [r y], m = [m1 m2] and
of
C =
c-( )
ρσισ
1) With
c-()
4.
C =
9
we obtain of = 4, o = 9 and pơ102 = -4. Thus
2) The transformation Z = 2X +Y, W = X – 2Y is written in matrix notation as
()-(: )G)-()
2
1
= A
W
The ditribution fz,w(z, w) is binormal with mean m' = mA', and covariance matrix C' =
ACA'. Hence
C' = ({
9 2
2
4 -4
2
1
1 -2
9
-2
2 56
The off-diagonal elements of C' are equal to po zow = COV(Z,W). Thus COV (Z, W) = 2.
3) Z will be Gaussian with variance o? = 9 and mean
%3D
mz = [ m1
m2
= 4
21
![4.27 Random variables X and Y are jointly Gaussian with
m = [1 2]
-4
4
C =
9.
1. Find the correlation coefficient between X and Y.
2. If Z = 2X + Y and W = X – 2Y, find COV(Z, W).
3. Find the PDF of Z.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa80eb220-b198-4f3d-81bb-2cbd72fe9990%2Ffba5fc90-9255-42b5-ad8b-53172751698d%2Ftspu9x5_processed.png&w=3840&q=75)
Transcribed Image Text:4.27 Random variables X and Y are jointly Gaussian with
m = [1 2]
-4
4
C =
9.
1. Find the correlation coefficient between X and Y.
2. If Z = 2X + Y and W = X – 2Y, find COV(Z, W).
3. Find the PDF of Z.
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