## Quick Reference

A theorem, given by Cochran in 1934, concerning sums of chi-squared variables. Let **Y** represent an *n*×1 vector of independent standard normal random variables and let **A**_{1}, **A**_{2},…, **A*** _{k}* be non-zero symmetric matrices such that , where

**Y**′ is the transpose of

**Y**. Write

*Q*

*=*

_{j}**Y**′

**A**

_{j}**Y**. Cochran's theorem, published in 1934, states that, if any one of the following three conditions is true, then so are the other two:

**A**

_{1},

**A**

_{2},…,

**A**

*sum to*

_{k}*n*.

*Q*

_{1},

*Q*

_{2},…,

*Q*

*has a chi-squared distribution.*

_{k}*Q*

_{1},

*Q*

_{2},…,

*Q*

*is independent of all the others.*

_{k}**A**_{1}, **A**_{2},…, **A*** _{k}* sum to

*n*.

*Q*_{1}, *Q*_{2},…, *Q** _{k}* has a chi-squared distribution.

*Q*_{1}, *Q*_{2},…, *Q** _{k}* is independent of all the others.

*Subjects:*
Probability and Statistics.

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