5 Expectation

5.1 Expectation \(\mathbb{E}[x]\)

\[\begin{aligned} \text{Law of total expectation:} \quad \mathbb{E}[X] &= \sum_j \mathbb{E}(E|F_j)\mathbb{P}(F_j)\\ \\ \text{Discrete:} \quad \mathbb{E}[X] &= \mu = \sum_{i=1}^k x_i p_i \\ \\ \text{Continuous:} \quad \mathbb{E}[X] &= \mu = \int_{-\infty}^{\infty} x f(x) dx \\ \end{aligned}\]

Conditional expectation:

\(\mathbb{E}[X] = \mathbb{E}(\mathbb{E}[X|Y])\)

\(\mathbb{E}[XY] = \mathbb{E}(\mathbb{E}[XY|Y]) = \mathbb{E}(Y\mathbb{E}[X|Y])\)

5.2 Variance \(\text{Var}[x]\)

where \(a\) is a constant:

\(Var(ax) = a^2 var(x)\)

\(Var(a) = 0\)

5.3 Covariance \(\text{Cov}[x,y]\)

\(\text{Cov}(X,Y) = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y]\)

5.4 Correlation Coeffecient \(\rho\)

\(\rho (x,y) = \frac{\text{Cov}(x,y)}{\sqrt{\text{Var}[X]}\sqrt{\text{Var}[Y]}}\)