Q on Bivariate normal

  • Thread starter autobot.d
  • Start date
  • #1
68
0

Homework Statement


Z is a 2x1 multivariate gaussian random vector, where [tex] Z = (X Y)^t [/tex], X,Y are real numbers, with mean zero and covariance matrix

[tex] \Gamma[/tex]
which is a 2x2 matrix whose entries are
[tex] \Gamma_{1,1} = 1[/tex]
[tex] \Gamma_{1,2} = \alpha [/tex]
[tex] \Gamma_{2,1} = \alpha [/tex]
[tex] \Gamma_{2,2} = 1 [/tex]

with [tex] | \alpha | < 1 [/tex]

a) Find the joint distribution of [tex] W_1 = X [/tex] and
[tex] W_2 = X+Y[/tex]

b) Find the conditional pdf of X+Y given X.




The Attempt at a Solution



I think I want to do a linear transformation to get a) but not sure how to attack the problem. Any help/references would be greatly appreciated. This is easy for independent gaussian variables but this is not the case here.

Thanks.
 

Answers and Replies

  • #2
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,722

Homework Statement


Z is a 2x1 multivariate gaussian random vector, where [tex] Z = (X Y)^t [/tex], X,Y are real numbers, with mean zero and covariance matrix

[tex] \Gamma[/tex]
which is a 2x2 matrix whose entries are
[tex] \Gamma_{1,1} = 1[/tex]
[tex] \Gamma_{1,2} = \alpha [/tex]
[tex] \Gamma_{2,1} = \alpha [/tex]
[tex] \Gamma_{2,2} = 1 [/tex]

with [tex] | \alpha | < 1 [/tex]

a) Find the joint distribution of [tex] W_1 = X [/tex] and
[tex] W_2 = X+Y[/tex]

b) Find the conditional pdf of X+Y given X.




The Attempt at a Solution



I think I want to do a linear transformation to get a) but not sure how to attack the problem. Any help/references would be greatly appreciated. This is easy for independent gaussian variables but this is not the case here.

Thanks.

Just use the bivariate moment-generating function
[tex] M(w_1,w_2) = E \exp(w_1 W_1 + w_2 W_2) = \int \int e^{w_1 v_1 + w_2 v_2} f_{W_1 W_2}(v_1,v_2) \: dv_1 \: dv_2.[/tex] See, eg.,
http://www.public.iastate.edu/~maitra/stat501/lectures/MultivariateNormalDistribution-II.pdf
 

Related Threads on Q on Bivariate normal

  • Last Post
Replies
9
Views
1K
Replies
6
Views
5K
Replies
2
Views
5K
Replies
8
Views
3K
Replies
3
Views
3K
  • Last Post
Replies
1
Views
827
  • Last Post
Replies
11
Views
2K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
4
Views
2K
Top