I've been revising some statistics lately, namely random variable distributions. My friend has given me some distributions he came up with and I'm trying to find the distributions of sum of variables etc.
He gave me 1/sqrt(1-x^2) and I'm pretty much stuck, because every way I know of doing this kind of problems leads me to an integral that wolfram doesn't even dare to touch.
Pic rel, I've tried the very standard method and ended up with integral that I have no idea how to solve and wolfram comes up with elliptic integrals. I've also tried to go through the cumulative distr. function but ended up later with integral of arcsin(t-x)/sqrt(1-x^2) that I can't solve either.
Any hints? Thanks.
Anyone? I haven't made any progress so far.
>>8119744
you can probably approximate it with some normal distribution for high enough n
>>8120218
Well, I've generated it numericaly and it's nowhere like normal distribution. More like two exponents mirrored and glued together with peak value in the middle.
>>8119744
Slightly confused what you're trying to find here. You've been given a density function of [math]1/\sqrt{1-x^2}[/math] and want to find the distribution of the transformation 2x? Or are you dealing with multivariate? I see you have both the 2x transformation written above, and a multivariate case written below.
>>8120320
That X+X thing is a mistake.
I want to find a distribution of a sum of independent variables of distribution f(x) each.
>>8120325
Have you considered using the moment generating function method? Independently distributed variables tend to work better with that.
>>8120330
I've never learned that, but I'll give it a look tomorrow. Thank you, looks promising.
>>8120335
It relies on your givens to be fairly well known so you can actually use the MGF, and off the top of my head I don't know what type of distribution that density is similar to or what MGF it would use.
>>8120337
So the MGF of this distribution seems to be, as Wolfram says, pi* J0(t), where J0(t) is a Bessel function, that would give MGF of Y = X1 + X2 equal to pi^2 * J0^2. I have no idea how to go back to the probability function.
>>8120375
That's the issue lol, relies on it working out to a recognizable MGF. Your work looks kosher up to the point where you have to integrate the g function for the marginal distribution of u. Did your friend bother working through the problem before actually giving you this, or did he just spew out a density to work with?
>>8120407
>or did he just spew out a density to work with?
Pretty much this.
>>8120413
Eh, fuck the final answer then as far as im concerned then. Your work is correct. You knew to arbitrarily map y to v, found the determinant of the jacobian matrix, worked out the bounds of the support, and attempted to integrate out the arbitrary mapped variable. If the only issue is some tedious integration, I think you got enough work done to say you know how to find a distribution using the multidimensional jacobian method.
Maybe there's a trick to integrating or something mechanical like that, but whatever.
>>8120425
Yeah, I think I'll leave it at that and go to bed. Thanks for help. Good night Anon.
