Book 3.3.1: Average vs Expected value

[dimitry-ishenko]

The two terms—average and expected value—are used somewhat interchangeably in the book (mainly chapters 3.1 and 3.2) even though they are, strictly speaking, two different things. For example, if I have 6-sided die, my expected value is always:

$$E[X] = 1 \cdot \frac 1 6 + 2 \cdot \frac 1 6 + 3 \cdot \frac 1 6 + 4 \cdot \frac 1 6 + 5 \cdot \frac 1 6 + 6 \cdot \frac 1 6 = 3.5$$

But, my average can be anything between 1 and 6. If I roll the die 10 times and get this sequence: 1 1 4 3 6 4 2 6 2 1, my average is:

$$\bar X = \frac {1 + 1 + 4 + 3 + 6 + 4 + 2 + 6 + 2 + 1} {10} = \frac {30} {10} = 3$$

The only thing that links the two values is the Law of Large Numbers, which states that average converges to the expected value as the number of samples increases.

Maybe some mention of the law can be made in the book and a few formulas tweaked where we equal the two.