Question

Here \(Q_{1}\) and \(Q_{2}\) are quadratic forms in observations of a random sample from \(N(0,1) .\) If \(Q_{1}\) and \(Q_{2}\) are independent and if \(Q_{1}+Q_{2}\) has a chi-square distribution, prove that \(Q_{1}\) and \(Q_{2}\) are chi-square variables.

Short answer

Using properties of chi-square distributions, it can be proven that both \(Q_{1}\) and \(Q_{2}\) are chi-square distributed.

Step-by-step solution

Understand the Assumptions

According to the problem, \(Q_{1}\) and \(Q_{2}\) are independent, and \(Q_{1}+Q_{2}\) is chi-square distributed. These are given conditions and do not need to be proven.

Apply property of chi-square distribution

The property of the chi-square distribution state that the sum of independent chi-square distributed random variables results in another chi-square distribution. Therefore, if \(Q_{1}+Q_{2}\) is chi-square distributed, and \(Q_{1}\) and \(Q_{2}\) are independent, then \(Q_{1}\) and \(Q_{2}\) must each be chi-square distributed by the property of additive chi-square distributions.

Conclusion

Based on the assumptions, the additive property of the chi-square distribution and the independence of \(Q_{1}\) and \(Q_{2}\), it can be concluded that \(Q_{1}\) and \(Q_{2}\) must be chi-square distributed.