📂Probability Theory

Expected Value Defined by Measure Theory

Definition ¹

Let us assume that a probability space $( \Omega , \mathcal{F} , P)$ is given. The $E(X)$, defined as follows for a random variable $X$, is referred to as the (mathematical) expected value of $X$. $$E(X) := \int_{\Omega} X d P$$

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• If you haven’t encountered measure theory yet, you can disregard the term probability space.

Explanation

The definition of expected value, however complex it might seem with measure theory involved, is indeed difficult to comprehend merely from a succinctly written formula. To make this more accessible, one might employ the following two theorems to transform it into a form we are more familiar with.

• [1] For a given random variable $X : \Omega \to \mathbb{R}$, $$\int_{\Omega} g( X ( \omega )) d P (\omega ) = \int_{\mathbb{R}} g(x) d P_{X} (x)$$
• [2] If the density $f_{X} , g : \mathbb{R}^{n} \to \mathbb{R}$ is integrable over the absolutely continuous $P_{X}$ defined at $\mathbb{R}^{n}$, then $$\int_{\mathbb{R}^{n}} g(x) d P_{X} (x) = \int_{\mathbb{R}^{n}} f_{X} (x) g(x) dx$$

Hence, the expected value of $g(X)$, when referred to as $n=1$ in [2], $$\begin{align*} E(g(X)) =& \int_{\Omega} g(X) d P \\ =& \int_{\Omega} g( X ( \omega )) d P (\omega ) \\ =& \int_{\mathbb{R}^{1}} g(x) d P_{X} (x) \\ =& \int_{\mathbb{R}} g(x) f_{X} (x) dx \end{align*}$$ This indicates that even within measure theory, $\displaystyle E(g(X)) = \int_{-\infty}^{\infty} g(x) f_{X} (x) dx$ is not merely accepted as a definition but can be derived. Especially in [1], if $g(x) = x$, it aligns with the introduced concept of expected value.

See Also

• Expected value as defined in Mathematical Statistics