Gaussian Random Walks: The N-step Distribution

This past evening, my former quiz bowl teammate turned co-worker (not to mention long-time friend) Michael Whittaker tossed a number of math problems my way. One in particular caught my interest:

Say you’ve got a normal random variable with mean zero and variance one. Draw a sample from that distribution. Now take that point you’ve just sampled and spawn a unit-variance normal distribution centered around it; draw a sample from this distribution. Continue this process $N$ times for arbitrary $N$. What is the distribution of the $N$th sample?

Michael told me that he and his friends at Cornell thought up this problem during a lull in a dinner-time conversation. The solution is after the break. A hint: It’s either simpler than it appears or, if you’ve done a bit of math, just as simple as it appears. read more »