pdf of a simple random variable calculated two different ways.. with two different answers so we have a random variable Y with a uniform pdf on the interval [0,1]. the question is, what is the pdf of W=Y^2.
method 1 using the transformation of variables formula:
$f_w=f_y(g^{-1}(y))\dfrac{d}{dy}g^{-1}(y)=\dfrac{1}{2 \sqrt{w}}$
method 2 treating W like a product distriubtion of $Y*Y$ (
http://en.wikipedia.org/wiki/Product_distribution)
$f_w = \int_{-\infty}^{\infty} \dfrac{1}{|y|}f_y(w/y)f_y(y)dy = \int_w^1 \dfrac{1}{|y|}dy = -ln(w)$
both have similar cdfs. i don't know which is right or why i got different answers.

The 'remark' above is on the right track, but has semantic ambiguities. 
The OP is basically asking why is the pdf of $Y^2$ different to the pdf of $Y*Y$?
An easy way to think of the difference is: ... How would you simulate $Y^2$ vs the product of two independent $Y$ random variables?


*

*To generate $Y^2$, you would generate a SINGLE pseudorandom drawing of random variable $Y$, and then square it. 

*To generate the product of two independent $Y$ random variables, you would generate TWO (different) pseudorandom drawings of $Y$, and obtain the product of them.
