Mean of a random variable

The mean of a random variable $X$ is $\mu_X = \sum x_i P(x_i)$ (mu-X, the sum of each value times its probability), taken across every value of a distribution whose probabilities total 1. Each value enters in proportion to how likely it is, which is what separates it from an ordinary average of the values. It is a parameter: one fixed number belonging to the distribution, settled before any data exist.

Let $X$ be the number of defective items in a box, with $P(0) = 0.5$, $P(1) = 0.3$ and $P(2) = 0.2$. Then $\mu_X = 0(0.5) + 1(0.3) + 2(0.2) = 0.7$ defects. The physical name for that number is the balance point. Put weights of 0.5, 0.3 and 0.2 at positions 0, 1 and 2 along a beam and it balances at 0.7, because the pull on the left is $0.5(0.7) = 0.35$ and the pull on the right is $0.3(0.3) + 0.2(1.3) = 0.09 + 0.26 = 0.35$.

"The mean is 0.7, so about 70 percent of boxes have a defect" is the misreading to kill. A mean is not a proportion, and it is not capped at 1; it only looks like a percentage here because the counts happen to be small. The share of boxes with at least one defect is $0.3 + 0.2 = 0.50$. Change the values to 0, 1 and 10 with those same probabilities and $\mu_X$ jumps to $0(0.5) + 1(0.3) + 10(0.2) = 2.3$ while that share stays at 0.50.

The distinction that matters downstream is $\mu_X$ against $\bar{x}$ (x-bar). Inspect 20 real boxes and their average will land somewhere near 0.7 without matching it; inspect 20,000 and it will sit closer. The mean of the distribution is the fixed number a sample mean converges toward, never a forecast for the next box.

This is topic 2.9 of Unit 2, Parameters of Random Variables.