Showing $E(X^n) = n!a$ using induction. Let $X$ be a random variable with $a = E(X)$, and suppose $X$ has the
same distribution as $U(Y + Z)$, where
$U$ is uniformly distributed over $(0, 1)$, $U, Y, Z$ are independent and $X, Y, Z$ have the same distribution.
How can we prove using induction that $E(X^n) = n!a^n$ for $n = 1, 2,...$.
Any hint would be appreciated.
 A: Use Strong induction (that means, suppose it holds for every $k<n+1$). Notice that
$$E(U^{n+1}(Y+Z)^{n+1})=E(U^{n+1})E\left(\sum _{k=0}^{n+1}\binom{n+1}{k}Y^kZ^{n+1-k}\right )=\frac{1}{n+2}\sum _{k=0}^{n+1}\binom{n+1}{k}E(Y^k)E(Z^{n-k+1})=\frac{1}{n+2}\left (E(Z^{n+1})+E(Y^{n+1})+\underbrace{\sum_{k=1}^{n}\binom{n+1}{k}E(Y^k)E(Z^{n-k+1})}_{\text{induction here}}\right )$$
$$E(X^{n+1})=\frac{1}{n+2}\left (2E(X^{n+1})\sum_{k=1}^{n}\binom{n+1}{k}k!a^k\cdot (n-k+1)!a^{n-k+1}\right )=\frac{2E(X^{n+1})+a^{n+1}(n+1)!\cdot n}{n+2},$$
solving for $E(X^{n+1})$ you get $E(X^{n+1})=(n+1)!a^{n+1}.$
A: One should do a strong induction:
The base case ($n=1$) is essentially the definition of $a$.
Take $n \geq 1$ and assume inductively that, for all $k = 1,\ldots,n$, we have
$$\mathbb E (X^k) = k!a^k.$$
Then
$$
\newcommand{\EE}{\mathbb E}
\begin{alignat}4
\EE (X^{n+1}) &= \EE\big(U^{n+1}(Y+Z)^{n+1}\big) &\qquad&
\\ &= \EE (U^{n+1})\EE\big((Y+Z)^{n+1}\big) &&\text{independence}
\\ &=  \frac1{n+2}\ \ \ \ \EE\big((Y+Z)^{n+1}\big) &&\text{hint in comments}
\\ &= \frac1{n+2}\sum_{k=0}^{n+1}\binom{n+1}k\EE(Y^kZ^{n+1-k}) &&\text{linearity of expect.}
\\ &= \frac1{n+2}\sum_{k=0}^{n+1}\binom{n+1}k\EE(Y^k)\EE(Z^{n+1-k}) &&\text{independence again}
\\ &= \frac1{n+2}\sum_{k=0}^{n+1}\binom{n+1}k\EE(X^k)\EE(X^{n+1-k}) &&\text{$X,Y,Z$ ident. distributed}
\\ &= \frac2{n+2}\EE(X^{n+1})+\frac1{n+2}\sum_{k=1}^{n}\binom{n+1}k\EE(X^k)\EE(X^{n+1-k})
\\ &= \frac2{n+2}\EE(X^{n+1})+\frac1{n+2}\sum_{k=1}^{n}\frac{(n+1)!}{k!(n+1-k)!}(k!a^k)\big((n+1-k)!a^{n-1+k}\big) && \text{induct. hypothesis}
\\ &= \frac2{n+2}\EE(X^{n+1}) + n!\cdot \frac n{n+2} a^{n+1} && \text{mass cancellation}
\end{alignat}
$$
and this last equality rearranges to
$$\frac{n}{n+2} \EE(X^{n+q})= \frac n {n+2}(n+1)!a^{n+1},$$
i.e. $$\EE(X^{n+1}) = (n+1)!a^{n+1},$$
completing the inductive step, and the proof.

Token remark: this implies the moment generating function no end!
$$ M_X:\lambda ↦ \EE (e^{\lambda X}) = \sum_{k=0}^∞ \frac{k!a^k\lambda^k}{k!}=\cdots=\frac 1{1-a\lambda}.$$
Up to choosing $a$, this fixes the distribution of $X$. Does anyone recognise if it corresponds to something named?
