Joint density of two vectors of multivariate normal random variables
up vote
1
down vote
favorite
If $bf{X}$ and $bf{Y}$ are dependent multivariate normal random variables, what is the joint density of $bf{X}$ and $bf{Y}$? Is it also multivariate normal?
probability matrices normal-distribution
edited Nov 21 at 10:43
p4sch
4,520217
asked Nov 21 at 5:27
shani
1208
add a comment |
up vote
1
down vote
favorite
If $bf{X}$ and $bf{Y}$ are dependent multivariate normal random variables, what is the joint density of $bf{X}$ and $bf{Y}$? Is it also multivariate normal?
probability matrices normal-distribution
edited Nov 21 at 10:43
p4sch
4,520217
asked Nov 21 at 5:27
shani
1208
2
On the other way round: If you know that they are independent, then they are jointly multivariate normal. If you only know they are dependent, then they are not necessary to be jointly multivariate normal in general.
– BGM
Nov 21 at 6:27
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
If $bf{X}$ and $bf{Y}$ are dependent multivariate normal random variables, what is the joint density of $bf{X}$ and $bf{Y}$? Is it also multivariate normal?
probability matrices normal-distribution
edited Nov 21 at 10:43
p4sch
4,520217
asked Nov 21 at 5:27
shani
1208
If $bf{X}$ and $bf{Y}$ are dependent multivariate normal random variables, what is the joint density of $bf{X}$ and $bf{Y}$? Is it also multivariate normal?
probability matrices normal-distribution
probability matrices normal-distribution
edited Nov 21 at 10:43
p4sch
4,520217
asked Nov 21 at 5:27
shani
1208
edited Nov 21 at 10:43
p4sch
4,520217
asked Nov 21 at 5:27
shani
1208
edited Nov 21 at 10:43
p4sch
4,520217
edited Nov 21 at 10:43
p4sch
4,520217
edited Nov 21 at 10:43
p4sch
4,520217
4,520217
asked Nov 21 at 5:27
shani
1208
asked Nov 21 at 5:27
shani
1208
asked Nov 21 at 5:27
shani
1208
1208
2
On the other way round: If you know that they are independent, then they are jointly multivariate normal. If you only know they are dependent, then they are not necessary to be jointly multivariate normal in general.
– BGM
Nov 21 at 6:27
add a comment |
2
On the other way round: If you know that they are independent, then they are jointly multivariate normal. If you only know they are dependent, then they are not necessary to be jointly multivariate normal in general.
– BGM
Nov 21 at 6:27
2
2
On the other way round: If you know that they are independent, then they are jointly multivariate normal. If you only know they are dependent, then they are not necessary to be jointly multivariate normal in general.
– BGM
Nov 21 at 6:27
On the other way round: If you know that they are independent, then they are jointly multivariate normal. If you only know they are dependent, then they are not necessary to be jointly multivariate normal in general.
– BGM
Nov 21 at 6:27
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
As already noted in the comments: In general, it is not true that $X$ and $Y$ have a jointly multivariate normal distribution. Here is one counterexample: Let $X sim mathcal{N}(0,1)$ and $Y$ independent of $X$ with $P(Y=1) =1/2$ and $P(Y=-1)=1/2$. Now set $Z=XY$.
- Show that $Z$ has also normal distribution with mean $0$ and variance $1$, i.e. $Z sim mathcal{N}(0,1)$.
$X$ and $Z$ are uncorrelated.
$X$ and $Z$ are not independent. (For example $P( |X| le t, |Z| >t) =0$, but $P(|X| le t) P(|Z|>t) ne 0$ for $t>0$.)
answered Nov 21 at 10:03
p4sch
4,520217
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
As already noted in the comments: In general, it is not true that $X$ and $Y$ have a jointly multivariate normal distribution. Here is one counterexample: Let $X sim mathcal{N}(0,1)$ and $Y$ independent of $X$ with $P(Y=1) =1/2$ and $P(Y=-1)=1/2$. Now set $Z=XY$.
- Show that $Z$ has also normal distribution with mean $0$ and variance $1$, i.e. $Z sim mathcal{N}(0,1)$.
$X$ and $Z$ are uncorrelated.
$X$ and $Z$ are not independent. (For example $P( |X| le t, |Z| >t) =0$, but $P(|X| le t) P(|Z|>t) ne 0$ for $t>0$.)
answered Nov 21 at 10:03
p4sch
4,520217
add a comment |
up vote
0
down vote
As already noted in the comments: In general, it is not true that $X$ and $Y$ have a jointly multivariate normal distribution. Here is one counterexample: Let $X sim mathcal{N}(0,1)$ and $Y$ independent of $X$ with $P(Y=1) =1/2$ and $P(Y=-1)=1/2$. Now set $Z=XY$.
- Show that $Z$ has also normal distribution with mean $0$ and variance $1$, i.e. $Z sim mathcal{N}(0,1)$.
$X$ and $Z$ are uncorrelated.
$X$ and $Z$ are not independent. (For example $P( |X| le t, |Z| >t) =0$, but $P(|X| le t) P(|Z|>t) ne 0$ for $t>0$.)
answered Nov 21 at 10:03
p4sch
4,520217
add a comment |
up vote
0
down vote
up vote
0
down vote
As already noted in the comments: In general, it is not true that $X$ and $Y$ have a jointly multivariate normal distribution. Here is one counterexample: Let $X sim mathcal{N}(0,1)$ and $Y$ independent of $X$ with $P(Y=1) =1/2$ and $P(Y=-1)=1/2$. Now set $Z=XY$.
- Show that $Z$ has also normal distribution with mean $0$ and variance $1$, i.e. $Z sim mathcal{N}(0,1)$.
$X$ and $Z$ are uncorrelated.
$X$ and $Z$ are not independent. (For example $P( |X| le t, |Z| >t) =0$, but $P(|X| le t) P(|Z|>t) ne 0$ for $t>0$.)
answered Nov 21 at 10:03
p4sch
4,520217
As already noted in the comments: In general, it is not true that $X$ and $Y$ have a jointly multivariate normal distribution. Here is one counterexample: Let $X sim mathcal{N}(0,1)$ and $Y$ independent of $X$ with $P(Y=1) =1/2$ and $P(Y=-1)=1/2$. Now set $Z=XY$.
- Show that $Z$ has also normal distribution with mean $0$ and variance $1$, i.e. $Z sim mathcal{N}(0,1)$.
$X$ and $Z$ are uncorrelated.
$X$ and $Z$ are not independent. (For example $P( |X| le t, |Z| >t) =0$, but $P(|X| le t) P(|Z|>t) ne 0$ for $t>0$.)
answered Nov 21 at 10:03
p4sch
4,520217
answered Nov 21 at 10:03
p4sch
4,520217
answered Nov 21 at 10:03
p4sch
4,520217
answered Nov 21 at 10:03
p4sch
4,520217
4,520217
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3007300%2fjoint-density-of-two-vectors-of-multivariate-normal-random-variables%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
2
On the other way round: If you know that they are independent, then they are jointly multivariate normal. If you only know they are dependent, then they are not necessary to be jointly multivariate normal in general.
– BGM
Nov 21 at 6:27