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If I have a variable X whose gaussian distribution is known and let f be a known function, is there a way to compute f(X) (i.e) the resulting gaussian distribution from this? Is the result actually a gaussian distribution?
[itex]E(f(z))= \int_{-\inf}^{\inf}{f(z)e^{-\frac{z^2}{2}}}dz[/itex]It won't be a gaussian distribution in general. For example if f(x) = x^{2} then you get what's called the chi-squared distribution with one degree of freedom (k degrees of freedom is adding the square of k gaussians). These are not gaussian (in fact it always has to give a positive number)
If you don't mind, can you write down the correct expression with the pi?That's correct except there should be a 1/sqrt(2pi) in there. In general if you have a probability density function p(x) for a random variable X, then
[tex] E(f(X)) = \int_{-\infty}^{\infty} f(x) p(x) dx [/tex]
In this case your p(x) is the Gaussian density.
[itex]E(f(z))= \frac{1}{sqrt(2\pi)}\int_{-\inf}^{\inf}{f(z)e^{-\frac{z^2}{2}}}dz[/itex]That's correct except there should be a 1/sqrt(2pi) in there. In general if you have a probability density function p(x) for a random variable X, then
[tex] E(f(X)) = \int_{-\infty}^{\infty} f(x) p(x) dx [/tex]
In this case your p(x) is the Gaussian density.