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A Conditional Distribution Problem

  1. May 4, 2008 #1
    The problem statement, all variables and given/known data
    Let [itex]Z_1, \ldots, Z_n[/itex] be independent standard normal random variables, and let

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

    What is the conditional distribution of [itex]S_n[/itex] given that [itex]S_k = y[/itex], for k = 1, ..., n?

    The attempt at a solution
    I know that [itex]S_j[/itex] 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 [itex]f_{S_n,S_k}(x,y)[/itex].
  2. jcsd
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