Gaussian Distribution Linear Combination of Variates

If $x$ is Normally Distributed with Mean $\mu$ and Variance $\sigma^2$, then a linear function of $x$,

$$y = ax+b,$$ (1)

is also Normally Distributed. The new distribution has Mean $a\mu+b$ and Variance $a^2\sigma^2$, as can be derived using the Moment-Generating Function

$$M(t) = \left\langle{e^{t(ax+b)}}\right\rangle{} = e^{tb}\left\langle{e^{atx}}\right\rangle{} = e^{tb}e^{\mu at + \sigma^2(at)^2/2}$$

$$= e^{tb+\mu at+ \sigma^2a^2t^2/2} = e^{(b+a\mu)t + a^2\sigma^2t^2/2},$$ (2)

which is of the standard form with

$$\mu' = b+a\mu$$ (3)

$$\sigma'^2 = a^2\sigma^2.$$ (4)

For a weighted sum of independent variables

$$y\equiv \sum_{i=1}^n a_ix_i,$$ (5)

the expectation is given by

$$M(t) = \left\langle{e^{yt}}\right\rangle{} = \left\langle{\mathop{\rm exp}\nolimits \left({t \sum_{i=1}^n a_ix_i}\right)}\right\rangle$$

$$= \langle e^{a_1tx_1}e^{a_2tx_2}\cdots e^{a_ntx_n}\rangle$$

$$= \prod_{i=1}^n\langle e^{a_itx_i}\rangle = \prod_{i=1}^n \mathop{\rm exp}\nolimits (a_i\mu_it + {\textstyle{1\over 2}}{a_i}^2{\sigma_i}^2t^2).$$ (6)

Setting this equal to

$$\mathop{\rm exp}\nolimits (\mu t + {\textstyle{1\over 2}}\sigma^2t^2)$$ (7)

gives

$$\mu \equiv \sum_{i=1}^n a_i\mu_i$$ (8)

$$\sigma^2 \equiv \sum_{i=1}^n {a_i}^2{\sigma_i}^2.$$ (9)

Therefore, the Mean and Variance of the weighted sums of $n$ Random Variables are their weighted sums.

If $x_i$ are Independent and Normally Distributed with Mean 0 and Variance $\sigma^2$, define

$$y_i\equiv \sum_j c_{ij}x_j,$$ (10)

where $c$ obeys the Orthogonality Condition

$$c_{ik}c_{jk}=\delta_{ij},$$ (11)

with $\delta_{ij}$ the Kronecker Delta. Then $y_i$ are also independent and normally distributed with Mean 0 and Variance $\sigma^2$.