1.5 Averaging 1.5.3 Averages of continuous variables 1.5.5 Problems

1.5.4 Properties of averages

When we were considering the average of a sum, eqn 1.15, we saw that the average of the sum was the sum of the average:

\langle\sum_{i=1}^{N}x_{i}\rangle=\sum_{i=1}^{N}\langle x_{i}\rangle

(1.18)

Or more simply $\langle x+y\rangle=\langle x\rangle+\langle y\rangle$ .

This is just because an average is itself a sum, and one overall multiplication, so the above equation is just rearranging the order you sum things in. Similarly if you multiply by any constant $c$

\langle cx\rangle=c\langle x\rangle

(1.19)

Another thing that you see a lot is the average of a constant. For example $\langle 2\rangle$ . What’s that? If every time you do the experiment you get $2$ . Then on average the result is 2. So for a constant $c$

\langle c\rangle=c

(1.20)

This means that the average of an average is just an average. I mean:

\langle\langle x\rangle\rangle=\langle x\rangle

(1.21)