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This is known as the truncated Stieltjes moment problem, and there is a necessary and sufficient condition taking the form of a semidefinite program. See Section 5 of the classic paper by Curto and Fi …

What can you do with the Kleisli category (of a probability monad) that you can't do with the Eilenberg-Moore category? The first paragraph of kirk sturtz's answer provides a good high-level summary …

Let $\mu$ be a probability measure on $[0,\infty)$ and $X_1, \dots, X_4 \sim \mu$ independent. Then what can be said about the probability that $X_1 + X_2 + X_3 < 2 X_4$? More precisely, what is the v …

In this answer, I will derive the following improved bounds: $$0.367 \approx \frac{208}{567} \le \sup_\mu \mathbf{P}[X_1 + X_2 + X_3 < 2 X_4] \le \frac{7}{15} \approx 0.467.$$ In the remarks at the en …

Here is a partial answer. As has been hinted at in the comments, one should expect that the space of functions under question strongly depends on the continuity assumption made. For example, the Rényi …

Let $X$ be a standard Borel space and $e : X \to X$ a Markov kernel. Suppose that $e$ is idempotent, that is $e \circ e = e$, or written out using the Chapman-Kolmogorov equation, $$e(A|x) = \int_X e( …

Let $(x_n)_{n \in \mathbb{N}}$ be a (deterministic) sequence of nonnegative reals, possibly even with $x_n \in \mathbb{N}$ if you prefer. Then we'd like to define the empirical measure of such a seque …

Since the idea of a probability monad is not a fully formal concept, there can be no fully formal answer to this question. Nevertheless, I will argue below that no nontrivial probability monad is a Ho …