Distribution of R from Sum of Squared RVs

[TranscendArcu]

Homework Statement

Let X,Y be two gaussian random variables that neither have the same variance nor the same mean necessarily, and both may be nonstandard. If I were to construct an RV of the form
R = √(X² + Y²), how would this RV be distributed?

The Attempt at a Solution

So I've given this a fair amount of thought. At first, I thought it might be distributed as a Rician, but that seems to require that both RVs have the same variance. Eventually I concluded that it might be Chi distributed. However, since I never divide by the standard deviation of either RV (as would be required to construct either a noncentral Chi or a noncentral Chi-Squared), this also fails to work. Does anyone have any ideas for finding the distribution of two RVs combined in this way?

 

[mighty2000]

I would approach this problem by first considering the properties of the RV R that we are trying to find the distribution for. We know that R is the square root of the sum of the squares of two Gaussian random variables, X and Y. This means that R is a non-negative continuous random variable.

Next, I would consider the moment-generating function (MGF) of R, which is defined as E[e^(tR)]. Using the properties of MGFs, we can see that the MGF of R would be the product of the MGFs of X and Y, since they are independent. This would give us an expression for the MGF of R in terms of the MGFs of X and Y.

From there, we can use the inverse transform method to find the probability density function (PDF) of R. This involves taking the inverse Laplace transform of the MGF of R. This can be a complex and lengthy process, but it would ultimately give us the distribution of R.

Another approach could be to use the characteristic function of R, which is the Fourier transform of the PDF. Again, we can use the independence of X and Y to find the characteristic function of R and then take the inverse Fourier transform to find the PDF.

In either case, the resulting distribution of R may not have a well-known name, but it would be a valid distribution that describes the random variable R. We could then use this distribution to calculate the mean, variance, and other properties of R, and also use it in any further calculations or analyses.

Overall, finding the distribution of R would require some mathematical calculations and possibly some approximations, but it is certainly possible to determine the distribution using the properties of R and the techniques of probability and statistics.