Re: Weighted sum of squared normals question

From: Ray Koopman (koopman_at_sfu.ca)
Date: 02/22/05 

Date: 22 Feb 2005 08:37:44 -0800

Ray Koopman wrote:
> Tom Diamond wrote:
>> Hello,
>>
>> I know that a sum of squared normals N(0,1) follows the chi-square
>> distribution with n degrees of freedom. What if the sum is weighted,
>> i.e. we have to add a1*x1^2 + a2*x2^2 + ... + aN*xN^2, where x1, x2,
>> ..., xN are normals N(0,1)?
>>
>> Tnx,
>>
>> Tom.
>
> It's approximately chi-square, with
>
> (a1 + a2 + ... + aN)^2
> df = ------------------------.
> a1^2 + a2^2 + ... + aN^2

That's not quite right. It's approximately *proportional*
to a chi-square variable. The constant of proportionality is

    a1^2 + a2^2 + ... + aN^2
c = ------------------------.
       a1 + a2 + ... + aN