A Conditional Distribution Problem

1. May 4, 2008

e(ho0n3

The problem statement, all variables and given/known data
Let $Z_1, \ldots, Z_n$ be independent standard normal random variables, and let

[tex]S_j = \sum_{i=1}^j Z_i[/itex]

What is the conditional distribution of $S_n$ given that $S_k = y$, for k = 1, ..., n?

The attempt at a solution
I know that $S_j$ is a normal random variable with mean 0 and variance j. The conditional density function is given by:

[tex]f_{S_n|S_k}(x,y) = \frac{f_{S_n,S_k}(x,y)}{f_{S_k}(y)}[/itex]

The denominator is easily found. All that's left to find is the numerator and with that I'll be able to find the conditional distribution function. This is where I'm stuck. I can't think of anything clever to determine $f_{S_n,S_k}(x,y)$.