$0 \leq Y \leq M$ random variable, $p > 1$.
Show that
$\mathbb{E}[Y^p] = \int_0^M py^{p-1}\mathbb{P}[Y \geq y] dy$
My attempt:
$\mathbb{E}[Y^p] = \int_0^{\infty} Y d\mathbb{P} = \int_0^{M} Y d\mathbb{P} $
Now I guess I have to use the Change of Variables Formula. I tried to do this but I didn't get the result I should.
By definition $$ \mathbb{E}[Y^p] = \int_0^{M} y^p f_Y(y)dy\ , $$ where $f_Y(y)$ is the pdf of $Y$, and thus equal to $f_Y(y)=\frac{d}{dy}\mathbb{P}[Y<y]=-\frac{d}{dy}\mathbb{P}[Y>y]$. Hence $$ \mathbb{E}[Y^p] = -\int_0^{M} y^p \frac{d}{dy}\mathbb{P}[Y>y]dy\ , $$ and using integration by parts $$ \mathbb{E}[Y^p] = -\left[y^p \mathbb{P}[Y>y]\right]\Big|_0^M + \int_0^M dy\ p y^{p-1}\mathbb{P}[Y>y]=\boxed{\int_0^M dy\ p y^{p-1}\mathbb{P}[Y>y]} $$