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What is the probability from a uniform probability measure (pg.37) on sample space $\left\{N(\theta,1)|\theta\in[0,1]\right\}$ that for some random variable $X$ in the sample space, the Expected-Value of $X$ is undefined or infinite?

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    $\begingroup$ It's either defined or undefined, infinite or not. There's no "probability" about it, as it's not random. $\endgroup$
    – jbowman
    Jan 22 at 21:33
  • $\begingroup$ Welcome to Cross Validated! Do you mean if $X$ is a particular random variable? What can you say about $X?$ $\endgroup$
    – Dave
    Jan 22 at 21:34
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    $\begingroup$ Randomly chosen from what distribution? Calculating a probability requires there to be a probability measure space. For instance, if the sample space is $\{N(\theta, 1)\vert \theta\in [0,1]\}$, with a uniform probability measure, then the probability is zero. If the sample space is just one Cauchy distribution with all density on that, then the probability is one. $\endgroup$
    – Dave
    Jan 22 at 21:49
  • $\begingroup$ @Dave I meant a uniform probability distribution on the sample space is $\left\{N(\theta,1)|\theta\in[0,1]\right\}$? Why is this zero? $\endgroup$
    – Arbuja
    Jan 22 at 22:06
  • $\begingroup$ Could you please explain what your notation means and what you mean by "sample space"? It would appear you are referring to a set of Normal distributions (with means between $0$ and $1,$ all of which are manifestly finite) but that is not what one would usually mean by a "sample space." Moreover, no "probability measure" has an expectation: expectations are properties of random variables. $\endgroup$
    – whuber ♦
    Jan 23 at 18:58

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If the sample space is $\{N(\theta,1)\vert \theta\in[0,1]\}$, then every point in that sample space is a distribution with a finite expectation (since they are normal). Consequently, for any random variable distributed according to an element of the sample space, that random variable will have a finite expectation, so there is zero probability of an element of this sample space having an undefined or infinite expectation, no matter the probability measure on the probability space, as one of the axioms of a measure (any measure, not just a probability measure) is that the empty set has measure zero.

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