I don't think there's any reason for it to have a nice distribution. After fooling around with it a bit, the following seem to be true:
*It has infinite variance (Not too hard to show, X/Y has infinite variance, and the other random variables just increase it)
*Pretty sure it has an infinite mean (Empirically, the sample-mean seems to be stationary, and theoretically, there's a bounding argument with a Cauchy that I'm not positive of.
Any specific questions about the distribution?
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