Ratio Distribution Question: C= (1/a)*exp(x/a), D= (1/b)*...

Hi there,

I am currently trying to solve a problem which is defined as follows:

C follows an exponential distribution with parameter a, D follows an exponential distribution with parameter b and a<b.

Thus, C= (1/a)*exp(x/a)
D= (1/b)*exp(x/b)

What is the distribution of G=C/D?

This should be, I would say: G = (b/a)*exp(x/b)/exp(x/a)

Am I correct so far? Well, next, on the basis of G, could the parameters a and b be estimated, or only a function of them? If so, what function?

I think only a function of them can be estimated, but I am thus far unable to find/define this function. Please help, thank you for your time.

Kind regards,

Thomas

Only a function of \(\displaystyle a/b\) can be estimated. Clean up your notation and find the distribution of the ratio of two independent exponential random variables.

Thank you for your reply. But that is just my problem, I have been searching through my books and googled a lot.

The only thing that comes close, as far as I can think of, is the F-distribution but I also read that this distribution is only used
for the ratio of two chi-square distributed variables. Therefore I do not know what the correct joint distribution is, in this case.

It is a multiple of an F distribution, with 2 degrees of freedom in numerator and denominator. Don't you know how to calculate the density of the ratio of two independent random variables?

You probably mean that I should be integrating the aformentioned formula resulting in the
ratio distribution?

Thus, G = (b/a)*exp(x/b)/exp(x/a) is correct? And all I now need to do is integrate it across x?

Thanks in advance.

You need to review change of variables and Jacobians. And start with the correct densities, which have a negative sign in the exponent.