Moments of Linear Combinations

If we have more than two random variables, $X_{1}, X_{2}, \dots, X_{n}$ , for constants $a$ :

E (\sum_{i = 1}^{n} a_{i} X_{i}) = \sum_{i = 1}^{n} a_{i} E (X_{i})

$E (X + 4 Y + 2 Z) = E (X) + 4 E (Y) + 2 E (Z)$

v a r (\sum_{i = 1}^{n} a_{i} X_{i}) = \sum_{i = 1}^{n} a_{i}^{2} v a r (X_{i}) + 2 \sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} a_{i} a_{j} \cdot c o v (X_{i} X_{j})

$V a r (X + 4 Y + 2 Z) = V a r (X) + 16 V a r (Y) + 4 V a r (Y) + 2 (1) (4) c o v (X, Y) + 2 (1) (2) c o v (X, Z) + 2 (4) (2) c o v (Y, Z)$

Covariances of Linear Combinations

Y_{1} = \sum_{i = 1}^{n} a_{i} X_{i} and Y_{2} = \sum_{i = 1}^{n} b_{i} X_{i}

then

c o v (Y_{1}, Y_{2}) = \sum_{i = 1}^{n} a_{i} b_{i} \cdot V a r (X_{i}) + \sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} (a_{i} b_{j} + a_{j} b_{i}) \cdot c o v (X_{i}, X_{j})

\begin{aligned} c o v (X + 4 Y + 2 Z, 3 X - Y - Z) = & (1) (3) V a r (X) + 4 (- 1) V a r (Y) + 2 (- 1) V a r (Z) + \\ + [(1) (- 1) c o v (X, Y) + 4 (3) c o v (X, Y)] + \\ + [(1) (- 1) c o v (X, Y) + (2) (3) c o v (X, Z)] + \\ + [4 (- 1) c o v (Y, Z) + 2 (- 1) c o v (Y, Z)] \end{aligned}