Firstly you forgot to multiply the
density f(xn=y) by 1/sqrt2pi. I think if you
obtained the density of the random variable X1,2=Xn1+Xn2 by
the convolution method, the problem no more posed , because for
Xn1+Xn2+Xn3=Xn1,2+Xn3=Xn1,2,3, and you have the
density of Xn1,2, the density of Xn3, you can calculate
there convolution, i.e the density of Xn1,2,3. If the calculation is
very difficult with the convolution (I think) you can
use the characteristic function. You calculate the function
characteristic of the variable Xn1 that one noted
psiXn1(t). As the two variables Xn1 and Xn2 are
i.i.d, then psiXn1+Xn2(t)=psiXn1(t)cdotpsiXn2(t)=(psiXn1(t))2 and so on for
variable Xn1+Xn2+...+Xnk we will have
psiXn1+Xn2+ldots+Xnk(t)=(psiXn1(t))k. Just well calculate psiXn1(t).
Friday, 22 August 2008
pr.probability - The density of x_1^n+x_2^n where x_i are Gaussian
at
11:48
Labels:
Mathematics

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