Re: Questions about two independent gamma variates
- From: Jack Tomsky <jtomsky@xxxxxxxxxxxxx>
- Date: Mon, 27 Feb 2006 12:34:27 EST
Dear all:
X1,, X2, Y are indenpendent normal variates
X1, X2,~ N(0,Sa)
Y~N(k,Sb)
then ( X1^2+X2^2) / (2Sa) is a gamma variate with a
DoF of 1
and (Y-k)^2/(2*Sb) is a gamma variate with
a DoF of 1/2
Question is what is the distribution of v^2 =X2^2+
X1^2+Y^2 ?
If we calculate is using integration, the P
(Y-k)^2/(2*Sb) goes to infinity
when Y approaches k.....
Is there any analytic form of this distribution?
Any thoughts are highly apprieciated
Regards
David
v^2 ~ Sa*Chisq(2,0) + Sb*Chisq(1,k^2/Sb)
where Chisq(2,0) is a central chi-square with two degrees of freedom and
Chisq(1,k^2/Sb) is a noncentral chi-square with one degree of freedom and noncentrality parameter k^2/Sb.
This is as far as you can go in terms of standard distributions. You can get an approximate distribution by fitting v^2 to a single multiple of a noncentral chi-square, A*Chisq(m,delta2), by matching up the first three moments and solving for A, m, and delta2.
Jack
.
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