Let $X$ be a random variable uniformly distributed over $(0,1)$. Assume a random variable $Y=X^n$, where $n$ is a fixed number. Please find the probability density function for the random variable $Y$; i.e., $f_Y (y)$?.

Could anyone point me in the right direction as far as formulas go for this? I'm kind of lost on how to start, and the book I have right now is not helping a single bit.

Thank you.


For these problems, it may be easier to manipulate the cumulative distribution function (CDF) first.

Notice that $Y$ is also supported on $(0,1)$.* Let's calculate its CDF. Fix $y \in (0,1)$. Then $$ \begin{eqnarray*} F_Y(y) = \Pr[Y \leqslant y] = \Pr[X^n \leqslant y] = \Pr[0 \leqslant X \leqslant y^{1/n}] = F_X(y^{1/n}) = y^{1/n}, \end{eqnarray*} $$ since $F_X(z) = z$ for $0 \leqslant z \leqslant 1$. Can you calculate the PDF of $Y$ from here?

*I am assuming that $n$ is a positive integer.


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