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section 2.2.7 of "Machine Learning: A Probabilistic Perspective by Kevin Patrick Murphy" gives this formula for expected value.

${\displaystyle \operatorname {E} [X]=\int_{\mathcal{X} }xf(x)\,dx.}$

There is a $\mathcal{X}$ at the bottom of integral symbol.

Assume $\mathcal{X}$ is a continuous real value random variable, since the book is discussing Machine Learning.

What does that $\mathcal{X}$ mean here? Does it mean "the domain of definition", "support" or something else?

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  • $\begingroup$ @Masacroso It's not the domain of $X$. In fact, $\mathbb X$ should be $\mathbb R$ if $X$ is a real continuous random variable. $\endgroup$ – amsmath Oct 14 '19 at 4:26
  • $\begingroup$ @amsmath you are right, my bad. Here $\Bbb X$ is the domain of $f$, not of $X$. $\endgroup$ – Masacroso Oct 14 '19 at 4:27
  • $\begingroup$ See here: en.wikipedia.org/wiki/… $\endgroup$ – amsmath Oct 14 '19 at 4:31
  • $\begingroup$ @amsmath Thank you! In your link, "the uniform distribution on the interval [0, ½] has probability density f(x) = 2 for 0 ≤ x ≤ ½ and f(x) = 0 elsewhere", is the interval [0, ½] the support of that uniform distribution, right? $\endgroup$ – JJJohn Oct 14 '19 at 4:45
  • $\begingroup$ @baojieqh Exactly! $\endgroup$ – amsmath Oct 14 '19 at 4:55
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$\Bbb X$ (or $\mathcal X$ after the edit) should indeed be the support for $X$, the domain over which the probability density function, $f$, returns non-zero values.

If $X$ is a real-valued random variable, then $\Bbb X\subseteq \Bbb R$.

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