# Students t-distribution Derivation

1. Oct 14, 2013

### taper100

I have attempted a derivation of the Students t-distribution probability distribution function in the attached pdf. I defined T to be Z/sqrt(W/v) where Z has standard normal distrubution and W has chi squared distribution with v degrees of freedom. I know that Z and W need to be independent, but I did not use this fact in my derivation. Can someone tell me where I went wrong in the derivation or where I unknowing used this fact in my derivation?

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• ###### StudentsTDerivation.pdf
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2. Oct 14, 2013

### Office_Shredder

Staff Emeritus
On the first line,
$$P(T\leq t | W = w) = P(\frac{Z}{c} \leq t )$$
assumes that Z and W are independent already, since you're saying that the probability that Z is smaller than some number is independent of what our measurement of W is.