Find PDF of $Y=-\frac{1}{2} \cdot \ln(X)$ given $f_X(x)=3x^2$

Question

I need to find the PDF of $Y=-\frac{1}{2} \cdot \ln(X)$ given $f_X(x)=3x^2$.

Can someone help me and give a clear explanation for what you should do for solving problems like this?

Answer 1

The only way I know how to solve problems like this is by considering the cumulative distribution function. I assume $X$ is distributed over $[0,1]$. Then:

$$F_X(x) : = P(X < x) = \int_0^x 3t^2\, dt = x^3$$

We compute the cdf of $Y$: $$P(Y<y) = P(-\tfrac12 \ln X < y) = P(\ln X>-2y) = P(X > e^{-2y}) = 1-F_X(e^{-2y}).$$

Can you complete it from here?