Help to prove $\sum_{i=1}^n(X_i-\overline{X})(\overline{X}-\mu)=0$.

Question

How can I show that $\sum_{i=1}^n(X_i-\overline{X})(\overline{X}-\mu)=0$? Anything that can point me in the right direction would help.

Answer 1

Assuming that $X_1, X_2, \dots, X_n$ is a random sample from a population with mean $\mu,$ where $\bar X = \frac 1 n \sum_{i=1}^n X_i$ (which you might have said), here is an outline of what you need to do:

$$\sum_{i=1}^n (X_i - \bar X)(\bar X - \mu) = \sum_{i=1}^n\left[\bar X X_i - \bar X^2 +\mu\bar X - \mu X_i\right]\\ = (n\bar X^2 - n\bar X^2) + (n\mu\bar X - n\mu\bar X) = 0.$$

You should give reasons for each step.