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.
Thanks for your replies. In my case .
Let me begin with the basis of my problem, which is the cosine law. In more detail I have the following:
The last term, i.e. is and is . I know that is Gaussian distributed. Now, since denotes distance it can take any values from . Also, and therefore I conclude that . On the other hand, is Gamma distributed with .
Therefore, I am faced with the subtraction of two random variables, . My aim of course is to find the density of but I approach the problem step by step. Having found , I then add the constant , which will not affect the variance of the distribution but only its first order moment. I have made use of the following theorem:
To proceed, I decompose the joint density function into the product of the conditional density of and the marginal . My intuition suggests that the conditional density should be Gaussian. This is due to being Gaussian given being Nakagami. Of course, will be Nakagami distributed if is Gamma.
Now, obviously (always positive), although this is not true for due to . Do you think that the support of as applied to this case will still be , so that the integral in (1) is evaluated accordingly? I think that the integration limits should be between and that the support of will be between . Please let me know your thoughts.
Thanks and apologies for the lengthy post.