Distributions summarized

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 §