Variance used in multivariable normal distribution

Question

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Why is variance: $$\sum_{i=1}^{m}\sum_{j=1}^{m}t_{i}t_{j}Cov(X_i,X_j)$$

Can someone derive this? Why dont they use the variance $$\sigma_{Y}^{2}=\sum_{i=1}^{n}\alpha_{i}^{2}\sigma_{i}^{2}$$

That is derived here:

https://onlinecourses.science.psu.edu/stat414/node/166

Answer 1

Recall that

$$ {\rm cov}(X, Y) = \mathbb{E}[(X - \mathbb{E}[X])(Y-\mathbb{E}[Y])] $$

so that

$$ {\rm cov}(X, X) = \mathbb{E}[(X - \mathbb{E}[X])^2] = {\rm Var}(X) $$

Therefore, if you call $Y = \sum_i t_i X_i$ then

\begin{eqnarray} {\rm Var}(Y) &=& {\rm cov}(Y, Y) = {\rm cov}\left( \sum_i t_i X_i, \sum_j t_j X_j\right) &=& \sum_{i,j}t_it_j{\rm cov}(X_i,X_j) \tag{1} \end{eqnarray}

With this being said, remember that for any two r.vs $X$ and $Y$

$$ {\rm Var}(X, Y) = {\rm Var}(X) + {\rm Var}(Y) + \color{red}{{\rm cov}(X,Y)} $$

Or in general

$$ {\rm Var}\left(\sum_i t_i X_i\right) = \sum_{i}t_i^2{\rm Var}(X_i) + \sum_{i\not= j}t_it_j \color{red}{{\rm cov}(X_i,X_j)} \tag{2} $$

which is exactly the same result as in (1). The problem with your last expression is that you are neglecting the covariance (red term in the equations above)