Moment and variance

• Definition: The kth moment of a continuous random variable $X$ is $E[X^k] = \displaystyle{ \int_{\Omega} } x^k f_X(x)$.

• Definition: The variance of a continuous random variable $X$ is $Var[X] = \mathbb{E}[(X - \mathbb{E}[X])^2] = \displaystyle{ \int\limits_{\omega} (x-\mu)^2 f_X(x) \;dx}$.

• The property $Var[X] = \mathbb{E}[X^2] - \mathbb{E}[X]^2$ still holds.

• Theorem: Let $g: \Omega \to \mathbb{R}$ be a function and $X$ be a continuous random variable. Then

$$\mathbb{E}[g(X)] = \int\limits_{\Omega} g(x) f_X(x)\; dx$$