Suppose I have a (possibly infinite) bag of coins with various weights. I select a coin and flip it $n$ times. Averaging over the choice of coins from the bag, there is some probability of seeing exactly $k$ heads, for $k=0,...,n$. Let $r_k$ be the probability of seeing exactly $k$ heads.

More formally, let $D$ be a probability distribution on the unit interval $[0,1]$, and let $h$ be a binomial random variable with parameters $n$ and $p$, where $p$ is drawn from $D$. Then marginalizing out $p$, the probability that $h=k$ is: $$r_k:=\int_{0}^{1} \binom{n}{k}p^k(1-p)^{n-k} dD(p)$$

Given $D$, we can (in principle) compute $(r_0,...,r_n)$. My question is: given $(r_0,...,r_n)$, does there exist some $D$ (i.e. bag of coins) that could have generated it? Or is the set of probabilities forbidden? More specifically, is there some finite procedure that I could follow to determine whether or not such a $D$ exists?

A few comments:

- Clearly we need $0\leq r_i\leq 1$ and $\sum_i r_i=1$.
- As a simple example of a forbidden configuration, take $n=2$ and $(r_0=0,r_1=1,r_2=0)$.
- Note that when $D$ exists, it is usually not unique.
- If we quantize the unit interval, e.g. if $D$ is supported on $\{0,\epsilon ,2\epsilon ,3\epsilon, ...,1\}$, we can express the quantized problem as a linear program. However, the LP fails to solve the original problem and also feels (to me) like overkill.

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