help with support of a function

**tecne** · February 2nd 2011, 06:56 AM

Hi,

I recently read your reply regarding the support of the function coming up from the addition of two random variables. I am referring to the following post:
sum of random variables - help to evaluate support of function

My problem is similar. I know that variable

is defined on

and also that variable

is defined on

. I am interested in the difference

. I am very confused however, since I think that the support of

and hence the integration limits should be between

. On the other hand, I think that the appropriate limits might be

in the convolution integral. Therefore, my problem is determining the integration limits of the convolution integral in order to find

.

Do you have any ideas? Your help will be greatly appreciated.

BR,

Alex

**mr fantastic** · February 2nd 2011, 11:34 AM

Originally Posted by tecne

Hi,

I recently read your reply regarding the support of the function coming up from the addition of two random variables. I am referring to the following post:
sum of random variables - help to evaluate support of function

My problem is similar. I know that variable

is defined on

and also that variable

is defined on

. I am interested in the difference

. I am very confused however, since I think that the support of

and hence the integration limits should be between

. On the other hand, I think that the appropriate limits might be

in the convolution integral. Therefore, my problem is determining the integration limits of the convolution integral in order to find

.

Do you have any ideas? Your help will be greatly appreciated.

BR,

Alex

If U = X - Y and

and

then the support of U will clearly be $\displaystyle -\infty < U < +\infty$ (in light of the extreme 'values' of X and Y, you should look at the extreme 'values' that it's possible for U to approach ....)

**tecne** · February 2nd 2011, 02:19 PM

Thanks for your reply. In my case $U=Y-X$ . However, I suspect that the same applies here and therefore $-\infty<U<\infty$ . I do not know the extreme values in advance. Do you think I should plug in these limits directly to the convolution integral?

Thanks again.

BR,

Alex

**mr fantastic** · February 2nd 2011, 07:12 PM

Originally Posted by tecne

Thanks for your reply. In my case $U=Y-X$ . However, I suspect that the same applies here and therefore $-\infty<U<\infty$ . I do not know the extreme values in advance. Do you think I should plug in these limits directly to the convolution integral?

Thanks again.

BR,

Alex

Please post the whole question.

**Volga** · February 3rd 2011, 04:40 AM

I thought,

$u=y-x$ , then $y=u+x$
$0<y<\infty$ translate that for u:
$0<u+x<\infty$
$-x<u<\infty$

So there should be at least one bound, line u=-x ???

Will wait for the full question...

**tecne** · February 3rd 2011, 11:50 AM

Thanks for your replies. In my case $Z=Y-X$ .

Let me begin with the basis of my problem, which is the cosine law. In more detail I have the following:

$r^2=y^2 + d^2 -2ydcos\gamma$ .

The last term, i.e. $2ydcos\gamma$ is $X$ and $y^2$ is $Y$ . I know that $X$ is Gaussian distributed. Now, since $y$ denotes distance it can take any values from $[0,\infty)$ . Also, $-1<cos\gamma<1$ and therefore I conclude that $-\infty<x<\infty$ . On the other hand, $Y$ is Gamma distributed with $0<y<\infty$ .

Therefore, I am faced with the subtraction of two random variables, $Z= Y-X$ . My aim of course is to find the density of $r^2$ but I approach the problem step by step. Having found $Z$ , I then add the constant $d^2$ , which will not affect the variance of the distribution but only its first order moment. I have made use of the following theorem:

$f_{Z}(z) = \int_{-\infty}^{\infty}f_{X,Y}(y-z,y)dy$ . (1)

To proceed, I decompose the joint density function into the product of the conditional density of $f(X \vert Y)$ and the marginal $f(Y)$ . My intuition suggests that the conditional density should be Gaussian. This is due to $X$ being Gaussian given $\sqrt{Y}$ being Nakagami. Of course, $Y$ will be Nakagami distributed if $Y^2$ is Gamma.

Now, obviously $0<y<\infty$ (always positive), although this is not true for $X$ due to $cos\gamma$ . Do you think that the support of $Z$ as applied to this case will still be $(-\infty,\infty)$ , so that the integral in (1) is evaluated accordingly? I think that the integration limits should be between $[0,\infty)$ and that the support of $Z$ will be between $(-\infty,\infty)$ . Please let me know your thoughts.

Thanks and apologies for the lengthy post.

BR,

Alex

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