(11) Let X,Y be independent random variables with Xx ~ Ezp(A1) and Y ~ Ezp(A2). Find the joint density of Z = X+Y, W = X – Y. The following answers are provided. Read the answers carefully before making your choice. (a) The inverse transformation is z = v (z, w) = (z+ w)/2, y= v½(z, w) = (z – w)/2. The jacobian of the transformation is the following determinant J: The bivariate density of Z, W is g(z, w) = A,dze(=+=)/2e¬Ag{=¬w)/2|J], if 플 > 0, 를 > . This is equivalent to saying that
(11) Let X,Y be independent random variables with Xx ~ Ezp(A1) and Y ~ Ezp(A2). Find the joint density of Z = X+Y, W = X – Y. The following answers are provided. Read the answers carefully before making your choice. (a) The inverse transformation is z = v (z, w) = (z+ w)/2, y= v½(z, w) = (z – w)/2. The jacobian of the transformation is the following determinant J: The bivariate density of Z, W is g(z, w) = A,dze(=+=)/2e¬Ag{=¬w)/2|J], if 플 > 0, 를 > . This is equivalent to saying that
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![(11) Let X, Y be independent random variables with X ~ Exp(A1) and Y - Erp(A2). Find the joint density of Z = X +Y,W = X – Y.
The following answers are provided. Read the answers carefully before making your choice.
(a) The inverse transformation is
I = v (z, w) = (z + w)/2, y = v2(z, w) = (z – w)/2.
The jacobian of the transformation is the following determinant
J =
=
The bivariate density of Z, W is
g(z, w) = A1A2e 1(=+w)/2e¬A2(z¬w)/2 |J],
if 쁘 > 0, > 0.
This is equivalent to saying that
if – z < w < z, 0 < z
g(z, w) =
if otherwise.
(b) The inverse transformation is
I = v (2, w) = (z + w)/2, y = v2 (z, w) = (z – w)/2.
The jacobian of the transformation is the following determinant
J =
=
The bivariate density of Z, W is
g(z, w)
if " > 0, " > 0.
This is equivalent to saying that
if – z< w < z, 0 < z
g(z, w) =
if otherwise.
(c) The inverse transformation is
I = v1 (z, w) = (z+ w)/2, y = v2(z, w) = (z – w)/2.
The jacobian of the transformation is the following determinant
J =
The bivariate density of Z, W is
g(2, w) = A1Aze A1(=+w)/2e¬A2{z=w)/2|J],
if > 0, 를 >0.
This is equivalent to saying that
if - 00 < w < 0o, 0< z
g(2, w) =
0,
if otherwise.
(d) The inverse transformation is
I = v1 (z, w) = (z + w)/2, y = v2(z, w) = (z – w)/2.
The jacobian of the transformation is the following determinant
J =
The bivariate density of Z, W is
g(2, w) = A1Aze A1(=+w)/2e-Ag(z=w)/2|J],
if " > 0," > 0.
This is equivalent to saying that
if – z < w < z, 0< z
g(2, w) =
0,
if otherwise.
(e) The inverse transformation is
z = v (z, w) = (z + w)/2, y = v2(z, w) = (z – w)/2.
The jacobian of the transformation is the following determinant
J =
The bivariate density of Z, W is
g(z, w) = A1Aze^1(=+w)/2eAg{==w)/2|J,
if > 0, > 0.
This is equivalent to saying that
if – z< w < z, 0 < z
g(z, w) =
if otherwise,
(a)
(b)
(c)
(d)
(e)
N/A
(vii- Select One)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F421db798-cca4-4efb-a6a9-3775d86708c7%2Fe96dacff-432d-4de9-a133-61b1a0c902ca%2Fa5c7f9o_processed.png&w=3840&q=75)
Transcribed Image Text:(11) Let X, Y be independent random variables with X ~ Exp(A1) and Y - Erp(A2). Find the joint density of Z = X +Y,W = X – Y.
The following answers are provided. Read the answers carefully before making your choice.
(a) The inverse transformation is
I = v (z, w) = (z + w)/2, y = v2(z, w) = (z – w)/2.
The jacobian of the transformation is the following determinant
J =
=
The bivariate density of Z, W is
g(z, w) = A1A2e 1(=+w)/2e¬A2(z¬w)/2 |J],
if 쁘 > 0, > 0.
This is equivalent to saying that
if – z < w < z, 0 < z
g(z, w) =
if otherwise.
(b) The inverse transformation is
I = v (2, w) = (z + w)/2, y = v2 (z, w) = (z – w)/2.
The jacobian of the transformation is the following determinant
J =
=
The bivariate density of Z, W is
g(z, w)
if " > 0, " > 0.
This is equivalent to saying that
if – z< w < z, 0 < z
g(z, w) =
if otherwise.
(c) The inverse transformation is
I = v1 (z, w) = (z+ w)/2, y = v2(z, w) = (z – w)/2.
The jacobian of the transformation is the following determinant
J =
The bivariate density of Z, W is
g(2, w) = A1Aze A1(=+w)/2e¬A2{z=w)/2|J],
if > 0, 를 >0.
This is equivalent to saying that
if - 00 < w < 0o, 0< z
g(2, w) =
0,
if otherwise.
(d) The inverse transformation is
I = v1 (z, w) = (z + w)/2, y = v2(z, w) = (z – w)/2.
The jacobian of the transformation is the following determinant
J =
The bivariate density of Z, W is
g(2, w) = A1Aze A1(=+w)/2e-Ag(z=w)/2|J],
if " > 0," > 0.
This is equivalent to saying that
if – z < w < z, 0< z
g(2, w) =
0,
if otherwise.
(e) The inverse transformation is
z = v (z, w) = (z + w)/2, y = v2(z, w) = (z – w)/2.
The jacobian of the transformation is the following determinant
J =
The bivariate density of Z, W is
g(z, w) = A1Aze^1(=+w)/2eAg{==w)/2|J,
if > 0, > 0.
This is equivalent to saying that
if – z< w < z, 0 < z
g(z, w) =
if otherwise,
(a)
(b)
(c)
(d)
(e)
N/A
(vii- Select One)
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