Expected Values

calendar_today March 31, 2026 article 88 words access_time 1 min replay Modified March 31, 2026

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Expected Value for Continuous RV

Definition 17.1 (Expected Value).

The expected value of a continuous random variable $X$ is

$$ E[X]=\int_{-\infty}^{\infty} x f_X(x)\,dx $$

Theorem 17.2 (Expected Value of a Function of a Continuous RV).

The expected value of a function $g(X)$ of random variable $X$ is

$$ E[g(X)]=\int_{-\infty}^{\infty} g(x) f_X(x)\,dx $$

Theorem 17.3 (Core Identities).

For any random variable $X$

$$ E[X-\mu_X]=0 $$

$$ E[aX+b]=aE[X]+b $$

$$ \mathrm{Var}[X]=E[X^2]-\mu_X^2 $$

$$ \mathrm{Var}[aX+b]=a^2\mathrm{Var}[X] $$

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