(a) Let (X,Y) follow the general bivariate normal distribution with pdf 1 (9(x) – h(x, y) + l(y) xp{- " ožoš(1 – p9))} 1 ON(x, y) еxp 2noxoY V1 – p² on (x, y) E R², where ux, HY E R, ox,oy > 0, –1
(a) Let (X,Y) follow the general bivariate normal distribution with pdf 1 (9(x) – h(x, y) + l(y) xp{- " ožoš(1 – p9))} 1 ON(x, y) еxp 2noxoY V1 – p² on (x, y) E R², where ux, HY E R, ox,oy > 0, –1
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![(a) Let (X,Y) follow the general bivariate normal distribution with pdf
1
1(9(x) – h(x, y) + l(y)
ON(x, y)
on (x, y) E R²,
exp
2no xoy V1– p²
where μΧ, μΥER, σχ, σΥ> 0,-1<ρ< 1,
9(x) – ož(x –
h(n, ) - 2ρσχ σΥ (x - μχ) (y - μY) and
e(y) – o (y – HY)².
Each row of the table below corresponds to a particular bivariate
normal distribution with pdf f(x,y). By matching the given pdf in each
row with the above general bivariate normal distribution, complete the
following table:
f(x, y)
HY
1
- exp(-2 – y)
1
exp(-22 – y/2)
1
2 exp{[-(x – 1)2 – (y + 1)*]}
1
1
exp {2(-² -v + vāry)}
1
1
1
For the rest of this question, suppose that X and Y follow a bivariate normal
distribution with the joint pdf
f(x, y) – - exp(-x² – y?) on (x, y) € R².
This corresponds to the case of the first row in the table above.
Consider the random variables Z - X+Y and W – 3X –Y.
(b) Show that the mean vector of Z and W is given by (0 0)".
(c) Find V(Z) and V(W).
(d) Verify that Cov(Z, W) – 1. Hence write down the covariance matrix of
Z and W.
(e) Are Z and W independent? Explain your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe77056ff-7cc2-4c89-898a-5d59d9e2cda1%2Fb033596c-827e-4d91-aa3f-4aaf64229ee9%2F99vzg47_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Let (X,Y) follow the general bivariate normal distribution with pdf
1
1(9(x) – h(x, y) + l(y)
ON(x, y)
on (x, y) E R²,
exp
2no xoy V1– p²
where μΧ, μΥER, σχ, σΥ> 0,-1<ρ< 1,
9(x) – ož(x –
h(n, ) - 2ρσχ σΥ (x - μχ) (y - μY) and
e(y) – o (y – HY)².
Each row of the table below corresponds to a particular bivariate
normal distribution with pdf f(x,y). By matching the given pdf in each
row with the above general bivariate normal distribution, complete the
following table:
f(x, y)
HY
1
- exp(-2 – y)
1
exp(-22 – y/2)
1
2 exp{[-(x – 1)2 – (y + 1)*]}
1
1
exp {2(-² -v + vāry)}
1
1
1
For the rest of this question, suppose that X and Y follow a bivariate normal
distribution with the joint pdf
f(x, y) – - exp(-x² – y?) on (x, y) € R².
This corresponds to the case of the first row in the table above.
Consider the random variables Z - X+Y and W – 3X –Y.
(b) Show that the mean vector of Z and W is given by (0 0)".
(c) Find V(Z) and V(W).
(d) Verify that Cov(Z, W) – 1. Hence write down the covariance matrix of
Z and W.
(e) Are Z and W independent? Explain your answer.
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