Distribution of the power of an exponential random variable 
Three students are working independently on their probability homework. They start at the same time. The times that they take to finish it are i.i.d. random variables $T_1$, $T_2$ , $T_3$ with $T_j^{1/\beta} \sim \operatorname{Expo}(\lambda)$, where β and λ are known positive constants.
Find the PDF of $T_1$

I simplify the notation by solving the equivalent problem: Let $X \sim \operatorname{Expo}(\lambda)$, and $Y=X^{\beta}$, find the PDF of $Y$.
by the change of variables formula,
\begin{align}
f_Y(y) &= f_X(g^{-1}(y)) * \lvert \frac{dg^{-1}(y)}{dy} \rvert \\
&=\lambda e^{-\lambda y^{1/\beta}}  \frac{y^{\frac{1-\beta}{\beta}}}{\beta} 
\quad y \geq0
\end{align}
Is this correct? I somehow could not find it to be true by using simulation.

That looks OK to me, and I derived it using the cumulative distribution method, so more or less independently of the change of variables formula. Also when I put $\lambda=1$ and $\beta=1/3$ and simulate this I get reasonably good agreement between the empirical distribution and the form for the density.

Histogram of Simulated Data and pdf Derived Frequency Polygon
