$\operatorname{Bin}{(n,U)}$, where $U$ is uniform on $(0,1)$ A question in my probability class:
Let $X$ have the binomial distribution $\operatorname{Bin}{(n,U)}$, where $U$ is uniform on $(0,1)$. Show that $X$ is uniformly distributed on $\{0,1,\dotsc, n\}$. (Hint: consider using one of the following tools: generating function, moment generating function, characteristic function.)
I'm having trouble understanding the question. Can someone first of all provide a reference for the meaning of $\operatorname{Bin}{(n,X)}$, where $X$ is a random variable? Then perhaps I can suss out how to proceed.
Update: My strategy is to use (for example) the characteristic function, and show it is the same as the characteristic function as if $X$ were uniform. I believe that random variables have the same distribution iff they have the same characteristic function, so that will solve it.
If I try to find the characteristic function $\varphi_{X}(t)$, I get
$$\varphi_X(t) = \mathbb{E}(e^{itX}) = \mathbb{E}(\mathbb{E}(e^{itX}\mid U))\\ = \mathbb{E} \left( \sum_{k=0}^n e^{itk} \binom{n}{k}u^k(1-u)^k \right)\\=\int_0^1  \left( \sum_{k=0}^n e^{itk} \binom{n}{k}u^k(1-u)^k \right)  du\\ = \sum_{k=0}^n e^{itk}\binom{n-1}{k}\frac1k.$$
If $X$ were uniformly distributed, it should have characteristic function 
$$\mathbb{E}(e^{itX}) = \sum_{k=0}^n \frac{1}{n}e^{itk},$$
so I must have made a mistake, since I understand that random variables have the same characteristic function iff they have the same distribution. (Unless there is some reason these two are equal that I'm not seeing.)

What you are looking for in a binomial distribution is repeating $n$ times an experiment with probability of success $p$, where both $n$ and $p$ are constant.
Imagine making determination of $p$ a part of the experiment. You first determine a uniform $p$ and then perform the experiment $n$ times and record the successes. $X$ is the number of those successes.
