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Given probability space $(\Omega, \mathcal{F}, P)$ and $P$-measurable function $X: \Omega \to \mathbb{R}$ called a random variable, define the expectation of $X$ to be $E[X]=\int_\Omega X(\omega)dP(\omega) = \int_\mathbb{R} xd\mu_X(x)$ where $\mu_X$ is a measure on $\mathbb{R}$ defined as $\mu_X(B)=P(X^{-1}(B))$.

I'm trying to learn some rigorous probability theory and I was wondering how to interpret the definition of the expectation. Thanks!

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  • $\begingroup$ It is the average of your random variable, namely random function, with respect to the probability measure associated to it. Actually your integral is still over the sample space but doesnt matter. $\endgroup$ Feb 25, 2015 at 3:21
  • $\begingroup$ I can't be the only one who finds it bizarre that someone would know what a measure is before they have some intuition about what expected value is, or know how to look that intuition up for themselves. $\endgroup$ Feb 25, 2015 at 3:59

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The best way is to forget about rigorous probability theory and look at elementary probability theory.

Another option is to consider the law of large numbers. Suppose that $X$ is a random variable such that $E[X^2] < \infty$. Let $X_1,X_2,X_3,\ldots$ be i.i.d. samples of $X$. Then almost surely $$ \lim_{n\to\infty} \frac{X_1+\cdots+X_n}{n} = E[X]. $$ That is, if you take the average of many copies of $X$, then you get something very close to $E[X]$, and in the limit, you get exactly $E[X]$. So the expectation is some sort of average value of your random variable.

If the random variable $X$ is discrete, then there is a simple explanation for the formula. Suppose that $X$ attains value in the domain $D$. Its expectation is $$ E[X] = \sum_{x \in D} \Pr[X = x]x. $$ What's the idea here? Suppose that you take $n$ samples of $X$. Roughly $\Pr[X=x]n$ of them are equal to $x$, and their contribution to the average is roughly $\Pr[X=x]x$. Summing over all $x$, we obtain the formula.

The formula for a general random variable is just a generalization of the simple formula for a discrete random variable. Don't forget that probability theory existed before the modern formalisms, and started its life in a discrete context.

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