Expected value

The expected value E[X] of a random variable X is the weighted average of the values x_i that the variable can take, multiplied by their probabilities of occurring: $$ E[X] = \sum_i x_i \cdot p(x_i) $$

The expected value might or might not be one of the possible outcomes of the random variable.

As a weighted average, the expected value is the number the phenomenon tends toward after many repetitions.

A practical example

Rolling a die leads to six possible outcomes.

$$ X = \ { \ 1 \ , \ 2 \ , \ 3 \ , \ 4 \ , \ 5 \ , \ 6 \ } $$

Each outcome has the same probability.

$$ p(1)=p(2)=p(3)=p(4)=p(5)=p(6)= \frac{1}{6} $$

The expected value of the random variable is E(X)=3.5.

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

Note: In this case, the expected value E(X)=3.5 is not one of the outcomes of the random variable X={1,2,3,4,5,6}. It's a limiting value that the average of the results will approach after many rolls of the die.

And so on.