# 'Finding' a normally distributed random variable

Let a random variable $Z$ have a standard normal distribution.

Suppose that we start at $0$. We 'walk' right, along the number line, till we reach $a$. We then turn around, walk back, past $0$, till we reach $-b$. We then turn and walk right again, past $0$ and $a$, until we reach $c$. The game continues like this, until we 'find' the random variable, by arriving at its position on the number line.

What is the optimal 'walking' strategy, as defined by the set of turning points $(a, -b, c, \ldots)$, to minimise the expectation of distance walked. And what is the expected distance walked for this optimal strategy?

• Nice question! I wonder what the close voters were thinking. Here's a reference and tutorial for typesetting math on this site. – joriki Jun 4 '16 at 7:40