Expectation, Variance and Covariance¶

This file gives the elementary information on Expectation, Variance and Covariance for the Random Variables.

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Expectation, Variance, Covariance¶

Expectation¶

Definition¶

If $X$ is a random variable with output space $H$ and with density $f(x)$ .

Then, $E[X]=\int_H x f(x) dx$

Linearity¶

Sum of random variables¶

If $X$ and $Y$ are two random variables with respective outputs spaces $H$ and $K$ , with respective densities $f$ and $g$ , with a joint density $f(x,y)$ and with a finite expectation.

We have $E[X+Y]=E[X]+E[Y]$

Idea of the proof¶

Let consider the function $q$ defined by $q(x,y)=x+y$

$E[q(X,Y)] = \int_{H}\int_{K} q(x,y) f(x,y)dx dy$

$\phantom{E[Z]} = \int_{H}\int_{K} (x+y) f(x,y)dx dy$

$\phantom{E[Z]} = \int_{H}\int_{K} x f(x,y)dx dy + \int_{H}\int_{K} y f(x,y)dx dy$

$\phantom{E[Z]} = \int_{H} x\int_{K} f(x,y)dy dx + \int_{K} y\int_{H} f(x,y)dx dy$

$\phantom{E[Z]} = \int_{H} xf(x) dx + \int_{K} y(y)g(y)dy$

$\phantom{E[Z]} = E[X]+E[Y]$

Product by a constant¶

We also have

$E[aX]=aE[X]$

Positivity¶

If $X$ is a positive random variable i.e $P(X\geq 0)=1$ then

$E[X]\geq 0$

Where $X$ is a positive random variable, if $E[X]=0$ , then

$P(X=0)=1$

Constant¶

$E[a] = a$

Covariance and Variance¶

Definition¶

$\textrm{Cov}(X,Y) = E[(X-E[X])(Y-E[Y])]$

Properties¶

Symmetry¶

$\textrm{Cov}(X,Y)=\textrm{Cov}(Y,X)$

Other expression¶

Sometimes more convenient

$\textrm{Cov}(X,Y) = E[XY] - E[X]E[Y]$

In particular

$\textrm{Var}(X) = \textrm{Cov}(X,X) = E[X^2] - E[X]^2$

Linearity¶

$\textrm{Cov}(aX+bY,Z) = a\textrm{Cov}(X,Z)+b\textrm{Cov}(Y,Z)$

Variance and covariance¶

$\textrm{Cov}(X,X) = E[(X-E[X])(X-E[X])]=\textrm{Var}(X)$

Note that the variance is always positive (as the expectation of a square of a random variable).

Covariance between a variable and a constant¶

$\textrm{Cov}(X,a) = 0$

Consequence :

$\textrm{Var}(a)=0$

Reciprocally, if a random variable has a variance equal to 0, then the variable is constant.

Variance of a linear combination¶

$\textrm{Var}\left(\sum_{i=1}^n\lambda_i Z_i\right) = \sum_{i=1}^n\sum_{j=1}^n \lambda_i\lambda_j\textrm{Cov}(Z_i,Z_j)$

Applications¶

$\textrm{Var}(aX) = a^2 \textrm{Var}(X)$

$\textrm{Var}(aX+bY)=a^2\textrm{Cov}(X,X)+2ab\textrm{Cov}(X,Y)+b^2\textrm{Cov}(Y,Y)$

$\textrm{Var}(aX-bY)=a^2\textrm{Cov}(X,X)-2ab\textrm{Cov}(X,Y)+b^2\textrm{Cov}(Y,Y)$

$\textrm{Var}(X+a) = \textrm{Var}(X)$

Covariance matrix¶

When we have a set of random variables $Z_1,\dots,Z_n$ .

For each pair $(k,l)$ , if we denote $c_{kl} = \textrm{Cov}(Z_k,Z_l)$

We can store the $c_{kl}$ 's in a matrix $\Sigma = \left[\begin{array}{ccc}c_{11} &\dots & c_{1n}\\ c_{21} & \dots & c_{2n}\\ \vdots & \ddots & \vdots\\ c_{n1} & \dots & c_{nn}\end{array}\right]$

$\Sigma$ is named the covariance matrix of the random vector $Z=\left[\begin{array}{c}Z_1\\ \vdots\\ Z_n\end{array}\right]$

Note that we can rewrite

$\textrm{Var}\left(\sum_{i=1}^n\lambda_iZ_i\right) = \lambda^T \Sigma \lambda$

where $\lambda =\left[\begin{array}{c}\lambda_1\\ \vdots\\\lambda_n\end{array}\right]$

and $^T$ designates the transposition

$\lambda^T =\left[\begin{array}{ccc}\lambda_1& \dots & \lambda_n\end{array}\right]$

Since a variance is always positive, the variance of any linear combination as to be positive. Therefore, a covariance matrix is always (semi-)positive definite, i.e

For each $\lambda$ $\lambda^T \Sigma \lambda\geq 0$

Cross-covariance matrix¶

Let consider two random vectors $X=(X_1,\dots,X_n)$ and $Y=(Y_1,\dots,Y_p)$ .

We can consider the cross-covariance matrix $\textrm{Cov}(X,Y)$ where element corresponding to the row $i$ and the column $j$ is $\textrm{Cov}(X_i,Y_j)$

If $A$ and $B$ are some matrices (of constants)

$\textrm{Cov}(AX,BY) = A\textrm{Cov}(X,Y)B^T$

Exercise¶

Suppose that we want to estimate a quantity modeled by a random variable $Z_0$ as a linear combination of known quanties $Z_1,\dots, Z_n$ stored in a vector $Z=\left[\begin{array}{c}Z_1\\ \vdots\\ Z_n\end{array}\right]$

We will denote $Z_0^\star = \sum_{i=1}^n \lambda_i Z_i = \lambda^T Z$ this (random) estimator.

We know the covariance matrix of the full vector $(Z_0,Z_1,\dots,Z_n)$ that we write with blocks for convenience:

$\left[\begin{array}{cc}\sigma_0^2 & c_0^T \\ c_0 & C\end{array}\right]$

where

$\sigma^2_0 = \textrm{Var}(Z_0)$
$c_0 = \textrm{Cov}(Z,Z_0)$
$C$ is the covariance matrix of $Z$ .

Compute the variance of the error $Z_0^\star-Z_0$

Solution¶

$\textrm{Var}(Z_0^\star-Z_0) = \textrm{Cov}(Z_0^\star-Z_0,Z_0^\star-Z_0)$

$\phantom{\textrm{Var}(Z_0^\star-Z_0)} = \textrm{Var}(Z_0) -2 \textrm{Cov}(Z_0^\star,Z_0) + \textrm{Var}(Z_0)$

$\phantom{\textrm{Var}(Z_0^\star-Z_0)} = \textrm{Var}(\lambda^TZ) -2 \textrm{Cov}(\lambda^T Z,Z_0) + \sigma_0^2$

$\phantom{\textrm{Var}(Z_0^\star-Z_0)} = \lambda^T\textrm{Var}(Z)\lambda -2 \lambda^T\textrm{Cov}( Z,Z_0) + \sigma_0^2$

$\phantom{\textrm{Var}(Z_0^\star-Z_0)} = \lambda^TC\lambda -2 \lambda^Tc_0 + \sigma_0^2$

Correlation coefficient¶

The covariance is a measure of the link between two variables. However it depends on the scale of each variable. To have a similar measure which is invariant by rescaling, we can use the correlation coefficient:

$\rho(X,Y)=\frac{\textrm{Cov}(X,Y)}{\sqrt{\textrm{Var}(X)\textrm{Var}(Y)}}$

When the correlation coefficient is equal to $1$ or $-1$ , we have

$Y=aX+b$

with

$a>0$ if $\rho(X,Y)=1$
$a<0$ if $\rho(X,Y)=-1$

Note that $\rho(X,Y)$ can be equal to $0$ even if the variables are strongly linked.

The usual example is a variable $X$ with a pair density ( $f(-x)=f(x)$ ) and $Y=X^2$ :

$\textrm{Cov}(X,Y)=\textrm{Cov}(X,X^2)=E[X^3]-E[X]E[X^2]=E[X^3]=\int_{\mathbb{R}} x^3f(x)dx =0$