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Hi, I am currently trying to calculate z^2 distribution when z is folloing (0,a) gaussian distribution

how can i calculate z^2 distribution?

Hernia Baby
on 26 Feb 2021

Do you want to calculate Chi-square distribution?

The distribution depends on the degree of freedom.

David Goodmanson
on 2 Mar 2021

Edited: David Goodmanson
on 2 Mar 2021

Hello JS

By gaussian I assume you mean the normal distribution, mean 0, std deviation 'a'

f(z) = N*exp(-(z/a)^2/2)

with N = 1/(sqrt(2)*sqrt(pi)*a)

y = z^2 has one sign while z can have either sign. Since the normal distribution is symmetric, to make life easier assume the initial distribution is one-sided, with z>=0 only, and double the height of the distribution to keep the normalization correct:

f(z) = 2*N*exp(-(z/a)^2/2) z>=0

Now that z >= 0 we can use z = sqrt(y). The the idea with pdfs is to find a function g(y) such that

f(z) dz = g(y) dy

with z = y^(1/2)

which means that dz = (1/2) y^(-1/2) dy

plug that in

2*N*exp(-(y/(2*a^2))) (1/2) y(-1/2) dy = g(y) dy

g(y) = N*exp(-(y/(2*a^2)))/sqrt(y)

which is a chi-squared probability distribution function (with 1 degree of freeedom) as noted by Jeff. This function has the disadvantage of being unbounded as y --> 0, although its area is still 1 as required.

If you meant y = (z/a)^2 instead of y = z^2, then the expression would be the standard chi-squared

g(y) = N*exp(-(y/2))/sqrt(y) N = 1/(sqrt(2)*sqrt(pi))

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