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Does there exist a distribution which is determined only by its non-integer moments? To put it another way, for $p \geq 0$, do there exist random variables $X$ and $Y$ supported on $(0, \infty)$ such that $\mathbb{E}[X^p] = \mathbb{E}[X^p]$ if and only if $p \not \in \mathbb{N}$?
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I recently tweeted something very silly (redundant information, I know). The tweet in question asked the following:
Let $X_0$ be supported on some nonempty $A \subseteq \mathbb{N}_{>0}$ with $\mathbb{P}(X_0 = k) = p_{0,k}$ and $\mathbb{E}[X_0] < \infty$. For each $n \geq 1$, recursively define $X_n$ on $\mathbb{N}_{>0}$ by $\mathbb{P}(X_n = k) = c_n \cdot k \cdot p_{n-1,k}$, where $c_n$ is a normalizing constant. Then, as $n \to \infty$
…then what?
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Back in 2020, I taught STA261 for the first time. The first part of that course deals with statistics (i.e., functions of random samples, not the subject as a whole!) and I chose to provide a light introduction to completeness because of how elegant the Lehmann-Scheffé theorem and related results in point estimation are down the road…