Suppose we have a normal distribution like
$ f(x) = \mathcal{N}(\mu = 30, \sigma^2=10) $
and we transform it to another function by multiplying it to
$ g(x) = 2x^2 $
the result would be:
$ f(x).g(x) = h(x) = \frac{2x^2}{\sqrt{2\pi}.\sigma}.e^{-\frac{(x-\mu)^2}{2.\sigma}} $
h(x) is not a normal distribution anymore, and it is not a p.d.f. either. The question is how we can transform it to a p.d.f?
After converting to a p.d.f, the result might not be a normal distribution. How we can find the $\mu$ for the new distribution? How about $\sigma^2$ ?