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Thread: help with support of a function

  1. #1
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    help with support of a function

    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
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  2. #2
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    Quote Originally Posted by tecne View Post
    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 $\displaystyle \displaystyle -\infty < U < +\infty$ (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 ....)
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  3. #3
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    Thanks for your reply. In my case $\displaystyle U=Y-X$. However, I suspect that the same applies here and therefore $\displaystyle -\infty<U<\infty$. 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
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  4. #4
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    Quote Originally Posted by tecne View Post
    Thanks for your reply. In my case $\displaystyle U=Y-X$. However, I suspect that the same applies here and therefore $\displaystyle -\infty<U<\infty$. 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.
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  5. #5
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    I thought,

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

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

    Will wait for the full question...
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  6. #6
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    Thanks for your replies. In my case $\displaystyle Z=Y-X$.

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

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

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

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

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

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

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


    Thanks and apologies for the lengthy post.

    BR,

    Alex
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