# Variance and covariance

## Introduction

### Variance

#### Definition

The variance of a random variable is given by:

\(Var(x)=E((x-E(x))^2)\)

\(Var(x)=E(x^2+E(x)^2-2xE(x))\)

\(Var(x)=E(x^2)+E(E(x)^2)-E(2xE(x))\)

\(Var(x)=E(x^2)+E(x)^2-2E(x)^2\)

\(Var(x)=E(x^2)-E(x)^2\)

#### Variance of a constant

\(Var(c)=E(c^2)-E(c)^2\)

\(Var(c)= c^2-c^2\)

\(Var(c)=0\)

#### Variance of multiple

\(Var(cx)=E((cx)^2)-E(cx)^2\)

\(Var(cx)=E(c^2x^2)-[\sum_i cx P(x_i)]^2\)

\(Var(cx)=c^2E(x^2)-c^2[\sum_i x P(x_i)]^2\)

\(Var(cx)=c^2[E(x^2)- E(x)^2]\)

\(Var(cx)=c^2Var(x)\)

#### Link between variance of expectation

\(E(x)^2+Var(x)=E(x)^2+E((x-E(x))^2)\)

\(E(x)^2+Var(x)=E(x)^2+E(x^2+E(x)^2-2xE(x))\)

\(E(x)^2+Var(x)=E(x)^2+E(x^2)+E(E(x)^2)-E(2xE(x))\)

\(E(x)^2+Var(x)=E(x)^2+E(x^2)+E(x)^2-2E(x)E(x))\)

\(E(x)^2+Var(x)=E(x^2)\)

#### Covariance

\(Var(x+y)=E((x+y)^2)-E(x+y)^2\)

\(Var(x+y)=E(x^2+y^2+2xy)-E(x+y)^2\)

\(Var(x+y)=E(x^2)+E(y^2)+E(2xy)-E(x+y)^2\)

\(Var(x+y)=E(x^2)+E(y^2)+E(2xy)-[E(x)+E(y)]^2\)

\(Var(x+y)=E(x^2)+E(y^2)+E(2xy)-E(x)^2-E(y)^2-2E(x)E(y)]\)

\(Var(x+y)=[E(x^2)-E(x)^2]+[E(y^2)-E(y)^2]+E(2xy)-2E(x)E(y)\)

\(Var(x+y)=Var(x) +Var(y)+2[E(xy)-E(x)E(y)]\)

We then define:

\(Cov(x,y):=E(xy)-E(x)E(y)\)

Noting that:

\(Cov(x,x)=E(xx)-E(x)E(x)\)

\(Cov(x,x)=Var(x) \)

So:

\(Var(x+y)=Var(x)+Var(y)+2Cov(x,y)\)

\(Var(x+y)=Cov(x,x)+Cov(x,y)+Cov(y,x)+Cov(y,y) \)

\(Cov(x,c)=E(xc)-E(x)E(c)\)

\(Cov(x,c)=cE(x)-cE(x)\)

\(Cov(x,c)=0\)

### Covariance matrix

With multiple events, covariance can be defined between each pair of events, including the event with itself.

The covariance between \(2\) variables is:

\(Cov(x_i,x_j):=E(x_ix_j)-E(x_i)E(x_j)\)

Which is equal to:

\(Cov(x_i,x_j)=E{[x_i-E(x_i)][x_j-E(x_j)]}\)

We can therefore generate a covariance matrix through:

\(\sum =E[(X-E[X])(X-E[X])^T]\)