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

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