If
is Normally Distributed with Mean
and Variance
, then
a linear function of
,
![\begin{displaymath}
y = ax+b,
\end{displaymath}](g_848.gif) |
(1) |
is also Normally Distributed. The new distribution has Mean
and
Variance
, as can be derived using the Moment-Generating Function
which is of the standard form with
![\begin{displaymath}
\mu' = b+a\mu
\end{displaymath}](g_853.gif) |
(3) |
![\begin{displaymath}
\sigma'^2 = a^2\sigma^2.
\end{displaymath}](g_854.gif) |
(4) |
For a weighted sum of independent variables
![\begin{displaymath}
y\equiv \sum_{i=1}^n a_ix_i,
\end{displaymath}](g_855.gif) |
(5) |
the expectation is given by
Setting this equal to
![\begin{displaymath}
\mathop{\rm exp}\nolimits (\mu t + {\textstyle{1\over 2}}\sigma^2t^2)
\end{displaymath}](g_859.gif) |
(7) |
gives
Therefore, the Mean and Variance of the weighted sums of
Random Variables
are their weighted sums.
If
are Independent and Normally Distributed with
Mean 0 and Variance
, define
![\begin{displaymath}
y_i\equiv \sum_j c_{ij}x_j,
\end{displaymath}](g_862.gif) |
(10) |
where
obeys the Orthogonality Condition
![\begin{displaymath}
c_{ik}c_{jk}=\delta_{ij},
\end{displaymath}](g_863.gif) |
(11) |
with
the Kronecker Delta. Then
are also independent and normally distributed with Mean 0
and Variance
.
© 1996-9 Eric W. Weisstein
1999-05-25