Suppose $X$ is an uniform random variable on the interval (0,1). Find the distribution and and density functions of $Y=X^n$ for $n\ge2$.
Since $n\ge2$ I say that $-\infty < y < \infty$. Where I am having trouble is solving $P(X^n\le y)$. My first reaction is to take the $nth$ root of both sides, but that really does not seem right. My other thought is to take the $\ln$ of each side. If the latter is correct would it be $P(X\le\ln(y))$, in which case you would have $1-F(\ln(y))$ for the distribution function?k After this point I now to then differentiate to get $f(y)$.