Bernoulli distribution §

The Bernoulli distribution is a discrete probability distribution for a random variable which takes the value 1 with probability $p$ and the value 0 with probability $q = 1 - p$.

$$f(k;p) = p^k(1-p)^{1-k} \text{ for } k \in \{0,1\}$$

Which is basiically, at k=1 (success) probability is $p$ and at k=0(failure) probability is $1-p$.

Mean §

The mean of the Bernoulli distribution is $p$.

Simple proof: $$E[X] = \sum_{k=0}^{1} k \cdot f(k;p) = 0 \cdot (1-p) + 1 \cdot (1 \cdot p = p) = p$$

Variance §

The variance of the Bernoulli distribution is $p(1-p)$.

Simple proof: $$E[X^2] = \sum_{k=0}^{1} k^2 \cdot f(k;p) = 0^2 \cdot (1-p) + 1^2 \cdot p = p$$

$$Var[X] = E[X^2] - E[X]^2 = p - p^2 = p(1-p)$$

Bivariate Gaussian distribution §

Have to add the formulae and the properties of the bivariate gaussian distribution §