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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*}

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