Thread: proving equation

let Y1,.....,Yn be random variables and let Y be the sample mean

show that sum(Yi-Y)^2 = sum(Yi^2)-nY^2 (the sum from 1 to n)

LHS=Sum(Yi^2+Y^2-2Yi*Y)=Sum(Yi^2)+Sum(Y^2)-2Sum(Yi*Y)

Help to complete

Originally Posted by Duke

let Y1,.....,Yn be random variables and let Y be the sample mean

show that sum(Yi-Y)^2 = sum(Yi^2)-nY^2 (the sum from 1 to n)

LHS=Sum(Yi^2+Y^2-2Yi*Y)=Sum(Yi^2)+Sum(Y^2)-2Sum(Yi*Y)

Help to complete

Sum(Y^2)=nY^2

2Sum(Yi*Y)=2Y*Sum(Yi)=2Y*nY=2nY^2

So, Sum(Yi-Y)^2=Sum(Yi^2)+nY^2-2nY^2=Sum(Yi^2)-nY^2

Thanks