I'm a math undergrad working on a psychology question: Assume a person subconsciously estimates a function $x(t)$ where $t$ is time since hearing a beep. However, with time, their estimate of time $t$ (via their internal clock) gets worse. I'm modeling this as a convolution with a Gaussian whose standard deviation increases linearly with time. Hence, the person's estimate of $x(t)$ will be
$E[x(t)] = \int_\tau \mathcal N(\tau; t, \sigma_t) x(\tau) d\tau = \int_\tau \frac{1}{\sqrt{(2 \pi \sigma_t^2)}} e^{-\frac{1}{2 \sigma_t^2} (t-\tau)^2}x(\tau) d\tau$.
and let's let $\sigma_t = \alpha t$. Now suppose we have access to the output - for example, $E[x(t)] = e^t$. How do we determine $x(t)$? Tried Fourier transforms (via change of variables, moved the time-dependence out of the Gaussian and into $x$, but didn't get far) and taking the Taylor expansion of $x(t)$ and applying the Gaussian on those (resulted in a mess). Any thoughts?
Thank you!
George
(P.S. first post - apologies for protocol transgressions!)