e Then what has to be true of x1 x2 x3 for sigma xi2 sigma

e) Then what has to be true of x1, x2, x3 for sigma xi^2 = (sigma xi)^2 to hold when n = 3?

Solution

e.

As

Sum( xi^2), n = 1 to 3 = x1^2 + x2^2 + x3^2

[Sum (xi), n = 1 to 3]^2 = (x1 + x2 + x3^2)^2 = x1^2 + + x2^2 + x3^2 + 2 x1 x2 + 2 x2 x3 + 2 x1 x3   

Then

x1^2 + x2^2 + x3^2 = x1^2 + + x2^2 + x3^2 + 2 x1 x2 + 2 x2 x3 + 2 x1 x3   

0 = 2 x1 x2 + 2 x2 x3 + 2 x1 x3   

0 = x1 x2 + x2 x3 + x1 x3 [ANSWER]

The equation above must hold for the condition to be true.