Probability Density Function of a function

Dear All,

let us say we have PDF(x) i.e. probability density function of variable x. How to find probability density function for f(x), PDF(f(x))?

Particularly

PDF(x) is

http://mathhelpforum.com/image/png;b...AASUVORK5CYII=

and f(x)=x^{2
Thanks in advance. }

Re: Probability Density Function of a function

Sorry expression for the PDF(x) did not appear it is PDF for Gaussian distribution.

Re: Probability Density Function of a function

I think that

PDF_f(x)(x)=const*PDF(inversf(x))

if PDF(x) is Gaussian and f(x)=x^2;

then PDF_x^2(x)=const*PDF(sqrt(x)) so the x should be replaced by sqrt(x) in Gaussian PDF in case of 0 mean. In case non 0 man x-mean should be replaced by sqrt(x-mean).

Re: Probability Density Function of a function

You are partially right. ). When can we say that and therefore ? This holds, in particular, when f increases monotonically. Then f is injective, so exists.

If X is non-negative, then because increases monotonically when . However, when X is normal, it can assume negative values. In this case, . If , this equals . In general, if , you can use the fact that where is the PDF of .

Re: Probability Density Function of a function

Dear emakarov,

thanks a lot, I would like to understand this in more ditails so could you pleas

suggest me abook where I can read about this issue?

Re: Probability Density Function of a function

Sorry, I have not been studying or teaching probability lately, so I don't know good textbooks. But this is very basic stuff: just the definition of PDF and properties of normal distribution. I think that every textbook on probability should cover this.