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Imagine a probability density function $f(x)$, defined for positive $x$, and let's note its $n$th non-centred moment $x_{n}$. The mean $x_{1}$ is fixed (and positive). How can I find $f(x)$ that ...

I have a question that seems very basic, and yet I have not managed to find an answer after probably several hours of Google-searching. Fix $0<a<b<\infty$, and let $\mathcal{P}_{[a,b]}$ be ...