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Mean of a random variable $X$ is its expectation.

I am interest in the new definition, mean of probability distributions

Let $p,q$ be probability density distributions. $0.5p+0.5q$ is the mean distribution.

How to generalize this definition of mean to uncountable infinite number of distributions $p$?

What is the proper textbook name for this "mean"?

dodo
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    mixtures are pretty great. Countable or uncountable infinity of distributions can be tackled with infinite series or integrals. – Galen Dec 04 '23 at 07:13
  • @Galen So I guess for my case it is "equally weighted mixture"? Do we need additional assumptions ot make sure that such mixture exist for uncountably infinite number of $p$? – dodo Dec 04 '23 at 07:27
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    A sum of an uncountable infinity is undefined: you need a measure-theoretic formulation. That has already been accomplished for you, because a mixture in this sense can be expressed as a convolution of a distribution with its mixing distribution. – whuber Dec 04 '23 at 14:25
  • @dodo See whuber's comment. – Galen Dec 04 '23 at 15:48
  • @whuber Many thanks! What I saw on the Wikipedia link is that only the distribution accompanied with an additional parameter, $\alpha$, can have infinite mixture as integration. Is that true? – dodo Dec 04 '23 at 19:07
  • As I wrote, uncountable sums are undefined in mathematics, so you need some kind of concept of integration for such sums. – whuber Dec 04 '23 at 21:12
  • @whuber I think you must be correct, though I am having a hard time understanding how convolution helps with the measure-theoretic definition needed. Is convolution here means the sum of random variable and the mixture of distribution? – dodo Dec 05 '23 at 12:01
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    Generally, you need a family of distributions $F(x;\theta)$ parameterized by a vector $\theta\in\Theta$ where $(\Theta,\mathrm d\mu)$ is a probability space so that you can form the mixture $$\tilde F(x)=\int_\Theta F(x;\theta),\mathrm d\mu(\theta).$$ Defining $f_x(\theta)=F(x;-\theta),$ that expression is the convolution $(f\star \mathrm d\mu)(0).$ – whuber Dec 05 '23 at 14:06

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