# Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions. Use this tag along with the tags (probability), (probability-theory) or (statistics).

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### How restrictive is the single-crossing property for a mean-preserving spread?

I am trying to prove some results based on a mean-preserving spread of a distribution, and they basically depend on the distributions satisfying a single-crossing property, like the figure here: ...
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### An inequality for a bisected "shifted quadrant" under a continuous symmetric bivariate distribution?

Suppose a bivariate probability distribution is continuous and has circular symmetry about the origin; i.e., the lines of constant density are concentric circles centered on the origin. Now consider ...
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### Density belongs to the Kullback-Leibler support of prior distribution, proof verification.

Suppose that we have some data generated from the a density $f_{0}$ with unknown true parameter $p_{0}$. Then $f$ is the density with parameter $p$ that we use to infer the true density $f_{0}$ or ...
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### Moments of two probability distributions match. Can the distributions have support differing by an interval?

Let me define the support of a probability distribution $f$ as the set of all locations $x$ where $f(x)>0$. I'll denote it $S(f)$. Suppose the distributions $f$ and $g$ have matching moments. Can ...
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I have two independent distributions $X$ and $Y$. $X$ is defined by the piecewise CDF F_X(x) = \begin{cases} F_X^1(x) & x \in (-\infty, a_1)\\ F_X^2(x) & x \in [a_1, a_2)\\ F_X^3(x) & x \...