Mean of a Random Variable vs Expected Value

The mean of a random variable is a weighted average in which each value counts in proportion to how likely it is, so $\mu_X = \sum x_i P(x_i)$ (mu-X, the sum of each value times its probability). Suppose the number of defective items in a box is 0 with probability $0.5$, 1 with probability $0.3$, and 2 with probability $0.2$. Then $\mu_X = 0(0.5) + 1(0.3) + 2(0.2) = 0.7$ defects. Notice that $0.7$ is not a value $X$ can ever take, which is fine: the mean is the balance point of the probability distribution, not a prediction for any single box.

The expected value is the mean you would see over many repetitions of the random process, and it need not be a value the variable can actually take. For example, a game that pays 2 dollars with probability 0.5 and nothing with probability 0.5 has expected value $2(0.5) + 0(0.5) = 1$ dollar. In general it is written $E(X) = \mu_X = \sum x_i \, p_i$ (E of X, the mean of X, sums each value times its probability). It is the balance point of the probability distribution.