Evaluating the average

We would like to solve the system

if

We compute the variance as follows :

We show that if

*
*

**Proof:***step 1: we show that for every constant *

Let us assume (by contradiction) that there exists

Hence :

Since for any m we have

Therefore, for any

Under the hypothesis that

*
*

*
So the variance could be expressed as :
*

we can substitute :

*Since ** and **, we have that **.
Remark : the condition *

We say that