Expected Value

144. Expected Value

144.1. Expected Value

The expected value $𝔼 [𝑋]$ is the probability-weighted average of a random variable — its long-run mean over many draws.

144.1.1. Discrete

Weight each value $𝑥_{𝑖}$ by its probability $𝑝_{𝑖}$ and sum:

𝔼 [𝑋] = \sum_{𝑖 = 1}^{𝑛} 𝑥_{𝑖} \cdot 𝑝_{𝑖}

Example

Fair die:

𝔼 [𝑋] = \sum_{𝑘 = 1}^{6} 𝑘 \cdot \frac{1}{6} = \frac{21}{6} = 3.5

144.1.2. Continuous

Swap the sum for an integral, and the mass $𝑝_{𝑖}$ for $𝑓 (𝑥) d 𝑥$ :

𝔼 [𝑋] = \int_{- \infty}^{\infty} 𝑥 \cdot 𝑓 (𝑥) d 𝑥

144.1.3. Partial (truncated) expectation

Average $𝑥$ over only a window $[𝑎, 𝑏]$ — same integrand, restricted limits:

\int_{𝑎}^{𝑏} 𝑥 \cdot 𝑓 (𝑥) d 𝑥 = 𝔼 [𝑋 \cdot 1_{{𝑎 \leq 𝑋 \leq 𝑏}}]

where $1_{{(\cdot)}}$ is the indicator (1 when the condition holds, 0 otherwise). This is not a mean — it is not renormalized, so it shrinks toward 0 as the window captures less probability.

144.1.4. Conditional expectation

Renormalize by the probability of the window — now it is a true average, given that $𝑋$ landed in $[𝑎, 𝑏]$ :

𝔼 [𝑋 | 𝑎 \leq 𝑋 \leq 𝑏] = \frac{\int_{𝑎}^{𝑏} 𝑥 \cdot 𝑓 (𝑥) d 𝑥}{𝑃 (𝑎 \leq 𝑋 \leq 𝑏)}

144.1.5. LOTUS

Law of the Unconscious Statistician. To average a function $𝑔 (𝑋)$ , integrate $𝑔$ against the density of $𝑋$ — no need to first find the distribution of $𝑔 (𝑋)$ :

𝔼 [𝑔 (𝑋)] = \int_{- \infty}^{\infty} 𝑔 (𝑥) \cdot 𝑓 (𝑥) d 𝑥