Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

Product of Two Gaussian PDFs

For the special case of two Gaussian probability densities,

\begin{eqnarray*}
x_1(t) &\isdef & \frac{1}{\sqrt{2\pi\sigma_1^2}}e^{-\frac{(t-\mu_1)^2}{2\sigma_1^2}}\\
x_2(t) &\isdef & \frac{1}{\sqrt{2\pi\sigma_2^2}}e^{-\frac{(t-\mu_2)^2}{2\sigma_2^2}}
\end{eqnarray*}

the product density has mean and variance given by

\begin{eqnarray*}
\mu &=&
\frac{\frac{\mu_1}{2\sigma_1^2} + \frac{\mu_2}{2\sigma_2^2}}{\frac{1}{2\sigma_1^2} + \frac{1}{2\sigma_2^2}}
\;\eqsp \;
\frac{\mu_1\sigma_2^2 + \mu_2\sigma_1^2}{\sigma_2^2 + \sigma_1^2}\\ [5pt]
\sigma^2 &=& \left. \sigma_1^2 \right\Vert \sigma_2^2 \;\isdefs \;
\frac{1}{\frac{1}{\sigma_1^2} + \frac{1}{\sigma_2^2}} \;\eqsp \;
\frac{\sigma_1^2\sigma_2^2}{\sigma_1^2 + \sigma_2^2}.
\end{eqnarray*}


Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2016-07-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA