andresthor
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Homework Statement
I'm trying to find the expected value of a probability distribution.
Homework Equations
\int_{-\infty}^\infty xP(x,t) = \int_{-\infty}^\infty x \frac{1}{\sqrt{4\pi Dt}}e^{-\frac{(x-dt)^2}{4Dt}}dx
The Attempt at a Solution
I expect the value to be something like $dt$ but then again I might be way off.
I've tried some different substitutions but have had no luck. One example of what I've been trying:
<br /> \begin{equation}<br /> \frac{1}{\sqrt{4\pi Dt}}\int_{-\infty}^\infty x e^{-\frac{(x-dt)^2}{4Dt}}dx=\left/ u=x-dt\right/ = \frac{1}{\sqrt{4\pi Dt}}\int_{-\infty}^\infty (u+dt) e^{-\frac{u^2}{4Dt}}du = \frac{1}{\sqrt{4\pi Dt}}\left(\int_{-\infty}^\infty u e^{-\frac{u^2}{4Dt}}du + dt \int_{-\infty}^\infty e^{-\frac{u^2}{4Dt}}du\right)<br /> \end{equation}<br />
But then I get nowhere.