Question

Suppose (X,Y,Z)^T has exponential distribution with mean vector (1,2,3)^T . Find the correlation matrix of (X,Y,Z)^T .

Answer 1

The probability density function (pdf) of an exponential distribution is

$f(x,\lambda)=\begin{cases} \lambda e^{-\lambda x} &x\ge0 \\ 0&x<0 \end{cases}$

correlation coefficients:

between X and Y:

$r_{XY}=\frac{cov(X,Y)}{\sigma_X \sigma_Y}$

for exponential distribution: $\sigma_X=E(X)$

$cov(X,Y)=E(XY)-E(X)E(Y)=E(XY)-2$

$r_{XY}=\frac{E(XY)-2}{2}$

between y and z:

$r_{YZ}=\frac{cov(Y,Z)}{\sigma_Y \sigma_Z}$

for exponential distribution: $\sigma_Y=E(Y),\ \sigma_Z=E(Z)$

$cov(Y,Z)=E(YZ)-E(Y)E(Z)=E(YZ)-6$

$r_{YZ}=\frac{E(YZ)-6}{6}$

between y and z:

$r_{XZ}=\frac{cov(X,Z)}{\sigma_X \sigma_Z}$

$cov(X,Z)=E(XZ)-E(X)E(Z)=E(XZ)-3$

$r_{XZ}=\frac{E(XZ)-3}{3}$

Correlation matrix: