what is the distribution $X_1+X_2^2+X_3^3+...+X_n^n$?

Question

Suppose that $X_1,X_2,...,X_n$ are i.i.d. Bernoulli$(p)$ random variables. What is the generating function of $X_1+X_2^2+X_3^3+...+X_n^n$. Can you identify the distribution?

---

I thought that because $X^2=X$ then I have a binomial distribution with parameter $n,p$ would that be a viable solution ?

Answer 1

Since $X_k^k = X_k$, $k\in\{1, 2, ..., n\}$ $$X_1+X_2^2+...+X_n^n = X_1+...+X_n$$ has Bernoulli distribution with parameters $n$ and $p$.