## Monday, November 26, 2012

### How misleading "expected value" can be

The expression "expected value" can be highly misleading.

I was just writing some research results and used the expression "expected value" in relation to a discrete random walk of the form

$x[n+1] = \left\{ \begin{array}{ll} x[n] + 1 & \qquad \text{with prob. } 1/2 \\ & \\ x[n] -1 & \qquad \text{with prob. } 1/2 \end{array}\right.$ .

This random walk is a martingale, so

$E\big[x[n+1]\big|x[n]\big] = x[n]$.

But from the above formula it's clear that it's never the case that $x[n+1] = x[n]$. Therefore, saying that $x[n+1]$'s expected value is $x[n]$ is misleading — in the sense that a large number of people may expect the event $x[n+1] = x[n]$ to occur rather frequently.

Mathematical language may share words with daily usage, but the meaning can be very different.

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Added Nov 27: In the random walk above, for any odd $k$, $x[n+k] \neq x[n]$. On the other hand, here's an example of a martingale where $x[n+1] = x[n]$ happens with probability $p$, just for illustration:

$x[n+1] = \left\{ \begin{array}{ll} x[n] + 1 & \qquad \text{with prob. } (1-p)/2 \\ & \\ x[n] & \qquad \text{with prob. } p \\ & \\ x[n] -1 & \qquad \text{with prob. } (1-p)/2 \end{array}\right.$ .

(Someone asked if it was possible to have such a martingale, which makes me fear for the future of the world. Also, I'm clearly going for popular appeal in this blog...)