PDF of transformed variable

Hi forum, any help appreciated!

**Notation**

is the cumulative distribution of x

is the probability density of x

**Problem**

Let X be a random variable with a uniform distribution on (-1,1)

Let [a is a constant between 0 and 1]

Find the pdf of Y.

**My attempt (ive highlighted in red where i assume my mistake is)**

i will use these results later:

Now try to find and then differentiate to get

* (apply inverse of g to both sides of inequality)*

* (simplify)*

* (the RHS is by definition the CDF of x)*

* (we know the functional form of the CDF of x is 0.5x+0.5, substitute)*

*Now we can differentiate (this is where i a assume my mistake is)*

use the rule that the derivative of is the reciprocal of the derivative of g.

* now, substitute the actual *

ive done some monte carlo simulations and im pretty sure my CDF(y) is correct but my PDF(y) isn't .... can anyone help? :)

Re: PDF of transformed variable

nevermind, figured it out.

i had got the calculus rule wrong, d'oh.