# What does $\mathcal{X}$ mean at the bottom of integral symbol $\operatorname {E} [X]=\int _{\mathcal{X} }xf(x)\,dx$?

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?

• @Masacroso It's not the domain of $X$. In fact, $\mathbb X$ should be $\mathbb R$ if $X$ is a real continuous random variable. – amsmath Oct 14 '19 at 4:26
• @amsmath you are right, my bad. Here $\Bbb X$ is the domain of $f$, not of $X$. – Masacroso Oct 14 '19 at 4:27
• See here: en.wikipedia.org/wiki/… – amsmath Oct 14 '19 at 4:31
• @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? – JJJohn Oct 14 '19 at 4:45
• @baojieqh Exactly! – amsmath Oct 14 '19 at 4:55

$$\Bbb X$$ (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 $$\Bbb X\subseteq \Bbb R$$.