91影视

Given that X₁, ... , X_n have a multivariate normal distribution.

$\mathrm{Cov}鈦�\left(X_i,X_j\right)=0$

The calculation is shown below,

$\mathrm{M}\left({\mathrm{t}}_1,{\mathrm{t}}_2,鈥�鈥�,{\mathrm{t}}_{\mathrm{n}}\right)=\exp 鈦�\left[\underset{i=1}{\overset{n}{鈭�}}鈥�t_i{渭}_i+\frac{1}{2}\underset{i=1}{\overset{n}{鈭�}}鈥�\underset{j=1}{\overset{n}{鈭�}}鈥�t_i{t}_j\mathrm{Cov}鈦�\left({\mathrm{X}}_{\mathrm{i}},{\mathrm{X}}_j\right)鈥�鈥�\left(\mathrm{I}\right)\right.$

Case I:

If ${\mathrm{X}}_1,{\mathrm{X}}_2,鈥�鈥�..{\mathrm{X}}_{\mathrm{n}}$are independent then the calculation will be,

$\mathrm{M}\left({\mathrm{t}}_1,{\mathrm{t}}_2,鈥�鈥�.{\mathrm{t}}_{\mathrm{n}}\right)={\mathrm{M}}_{{\mathrm{X}}_1}\left({\mathrm{t}}_1\right)鈰�{\mathrm{M}}_{{\mathrm{X}}_2}\left({\mathrm{t}}_2\right)鈥�鈥�..{\mathrm{M}}_{{\mathrm{X}}_{\mathrm{n}}}\left({\mathrm{t}}_{\mathrm{n}}\right)$

$=\exp 鈦�\left[{渭}_1{t}_1+\frac{1}{2}{蟽}_1^2{t}_1^2\right]\exp 鈦�\left[{渭}_2{t}_2+\frac{1}{2}{蟽}_2^2{t}_2^2\right]鈥�鈥�鈥�..\exp 鈦�\left[{渭}_n{t}_n+\frac{1}{2}{蟽}_n^2{t}_n^2\right]鈥�鈥�..\text{(II)}$

So from (I) and (II) we can see that both the expressions are equal when $X_1,X_2,鈥�.,X_n$ are independent if$Cov\left({\mathrm{X}}_{\mathrm{i}},{\mathrm{X}}_{\mathrm{j}}\right)=0;i鈮�j$

Case II:

If $\mathrm{Cov}鈦�\left({\mathrm{X}}_{\mathrm{i}},{\mathrm{X}}_{\mathrm{j}}\right)=0;i鈮�j$

Then

$\mathrm{M}\left({\mathrm{t}}_1,{\mathrm{t}}_2,鈥�.,{\mathrm{t}}_{\mathrm{n}}\right)=\exp 鈦�\left[\underset{i=1}{\overset{n}{鈭�}}鈥�{渭}_i{t}_i+\frac{1}{2}\underset{i=1}{\overset{n}{鈭�}}鈥�t_i^2\mathrm{Var}鈦�\left({\mathrm{X}}_{\mathrm{i}}\right)\right]$

$=\exp 鈦�\left[{渭}_1{t}_1+\frac{1}{2}{蟽}_1^2{t}_1^2\right]\exp 鈦�\left[{渭}_2{t}_2+\frac{1}{2}{蟽}_2^2{t}_2^2\right]鈥�鈥�鈥�鈥�\exp 鈦�\left[{渭}_n{t}_n+\frac{1}{2}{蟽}_n^2{t}_n^2\right]$

$鈬�{\mathrm{X}}_1,{\mathrm{X}}_2,鈥�鈥�{\mathrm{X}}_n$are independent Random variable

$\mathrm{Cov}鈦�\left({\mathrm{X}}_{\mathrm{i}},{\mathrm{X}}_{\mathrm{j}}\right)=0\mathrm{鈭赌}i鈮�j$

Therefore it has been shown that$X_1,鈥�,X_n$are independent random variables if$\mathrm{Cov}鈦�\left({\mathrm{X}}_{\mathrm{i}},{\mathrm{X}}_{\mathrm{j}}\right)=0$when$i鈮�j$.