Problem Suppose that X and Y are distributed bivariate normal with density given in Equation (17.38). Equation no. 17.38 a. Show that the density of Y given X = x can be written as х ехр 2ToxoyV1 – påY 8xу(х, у) — σχ HY|x fy[x=xV) = (x- Px 2рху 2т OYX {[(4):Grate where σγΧ= νσγ(1- ρy) and μγχ μγ- (σχΥ/σ) (-μ ). [Hint: Use the definition of the conditional probability density fyjx=xV) = [gx,Y(x, y)]/[fx(x)], where gx,y is the joint density of X and Y, and fx is the marginal density of X.] b. Use the result in (a) to show that Y|X = x ~ N(µy|x, oỷx). c. Use the result in (b) to show that E(Y|X = x) = a + bx for suitably chosen constants a and b.
Problem Suppose that X and Y are distributed bivariate normal with density given in Equation (17.38). Equation no. 17.38 a. Show that the density of Y given X = x can be written as х ехр 2ToxoyV1 – påY 8xу(х, у) — σχ HY|x fy[x=xV) = (x- Px 2рху 2т OYX {[(4):Grate where σγΧ= νσγ(1- ρy) and μγχ μγ- (σχΥ/σ) (-μ ). [Hint: Use the definition of the conditional probability density fyjx=xV) = [gx,Y(x, y)]/[fx(x)], where gx,y is the joint density of X and Y, and fx is the marginal density of X.] b. Use the result in (a) to show that Y|X = x ~ N(µy|x, oỷx). c. Use the result in (b) to show that E(Y|X = x) = a + bx for suitably chosen constants a and b.
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Question
Solve the Problem with Equaion 17.38
![Problem
Suppose that X and Y are distributed bivariate normal with density given
in Equation (17.38).
Equation no. 17.38
a. Show that the density of Y given X = x can be written as
х ехр
2ToxoyV1 – påY
8xу(х, у) —
σχ
HY|x
fy[x=xV) =
(x- Px
2рху
2т
OYX
{[(4):Grate
where σγΧ= νσγ(1- ρy) and μγχ μγ- (σχΥ/σ) (-μ ).
[Hint: Use the definition of the conditional probability density
fyjx=xV) = [gx,Y(x, y)]/[fx(x)], where gx,y is the joint density of X
and Y, and fx is the marginal density of X.]
b. Use the result in (a) to show that Y|X = x ~ N(µy|x, oỷx).
c. Use the result in (b) to show that E(Y|X = x) = a + bx for suitably
chosen constants a and b.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa3538d79-dd30-421a-8ca5-873dec048f25%2F914fdb88-0b3e-41dc-91ae-8ec243a44a3c%2F7u0wfz.png&w=3840&q=75)
Transcribed Image Text:Problem
Suppose that X and Y are distributed bivariate normal with density given
in Equation (17.38).
Equation no. 17.38
a. Show that the density of Y given X = x can be written as
х ехр
2ToxoyV1 – påY
8xу(х, у) —
σχ
HY|x
fy[x=xV) =
(x- Px
2рху
2т
OYX
{[(4):Grate
where σγΧ= νσγ(1- ρy) and μγχ μγ- (σχΥ/σ) (-μ ).
[Hint: Use the definition of the conditional probability density
fyjx=xV) = [gx,Y(x, y)]/[fx(x)], where gx,y is the joint density of X
and Y, and fx is the marginal density of X.]
b. Use the result in (a) to show that Y|X = x ~ N(µy|x, oỷx).
c. Use the result in (b) to show that E(Y|X = x) = a + bx for suitably
chosen constants a and b.
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