What does it mean to raise a distribution to a power?

Question

What does it mean in statistics if we raise distributions to the powers? Like if $X$ is uniformly distributed on $[0,1]$ then what is the distribution of $X^3$ or the expected value of $X^3$?

Answer 1

Let $Y=X^3$. Then $Y$ is also a random variable. The cumulative distribution function of $Y$ is given by $$ F_Y(y)=\Pr\{X^3\le y\}=\Pr\{X\le y^{1/3}\}=\int_0^{y^{1/3}}\mathrm dx=y^{1/3} $$ and the density function of $Y$ is given by $$ f_Y(y)=\frac13y^{-2/3} $$ for $y\in[0,1]$.

The expected value $\operatorname EY$ is given by $$ \operatorname EY=\frac13\int_0^1y\cdot y^{-2/3}\mathrm dy=\frac13\cdot\frac34=\frac14. $$