Re: Distribution of a transformed Gaussian variable.

beliavsky_at_aol.com
Date: 01/03/05 

Date: 3 Jan 2005 12:42:09 -0800

Fredrik Hekland wrote:
>I'm trying to find the distribution resulting from taking the square
>root of the absolute value of a Normally distributed variable. More
>precisely, p_z(z) when
>z = sign(n) * sqrt( constant*abs(n) ), n ~ N(0,sigma_n^2).
>Not having any statistical textbooks at my current location (and not
>being a statistics wizard), I'm unable to calculate the resulting pdf.
>Is it a complex task to determine this distribution?
>Simulations indicate that the resulting pdf is close to a Laplace
>distribution (double exponential). Does this seem reasonable?

The absolute value of a standardized normal variate follows the
half-normal distribution, described at
http://mathworld.wolfram.com/Half-NormalDistribution.html , with
distribution (2/pi) * exp(-x^2/pi). I think the square root of a
variate following this distribution has a density proportional to
sqrt(x)*exp(-x/pi) , for x > 0, which has the same decay as a Laplace
on the right side but not on the left. Please check this before using
it.

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