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Homework Statement
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].
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].