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

which is of the standard form with
\mu' = b+a\mu
\end{displaymath} (3)

\sigma'^2 = a^2\sigma^2.
\end{displaymath} (4)

For a weighted sum of independent variables
y\equiv \sum_{i=1}^n a_ix_i,
\end{displaymath} (5)

the expectation is given by
$\displaystyle M(t)$ $\textstyle =$ $\displaystyle \left\langle{e^{yt}}\right\rangle{} = \left\langle{\mathop{\rm exp}\nolimits \left({t \sum_{i=1}^n a_ix_i}\right)}\right\rangle$  
  $\textstyle =$ $\displaystyle \langle e^{a_1tx_1}e^{a_2tx_2}\cdots e^{a_ntx_n}\rangle$  
  $\textstyle =$ $\displaystyle \prod_{i=1}^n\langle e^{a_itx_i}\rangle = \prod_{i=1}^n \mathop{\rm exp}\nolimits (a_i\mu_it + {\textstyle{1\over 2}}{a_i}^2{\sigma_i}^2t^2).$ (6)

Setting this equal to
\mathop{\rm exp}\nolimits (\mu t + {\textstyle{1\over 2}}\sigma^2t^2)
\end{displaymath} (7)

$\displaystyle \mu$ $\textstyle \equiv$ $\displaystyle \sum_{i=1}^n a_i\mu_i$ (8)
$\displaystyle \sigma^2$ $\textstyle \equiv$ $\displaystyle \sum_{i=1}^n {a_i}^2{\sigma_i}^2.$ (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,
\end{displaymath} (10)

where $c$ obeys the Orthogonality Condition
\end{displaymath} (11)

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

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© 1996-9 Eric W. Weisstein