The mean, or average, or expected value of a random process is the sum of the possible outcomes times the probability of the outcome. |
For example, tossing two coins the possible outcomes are HH, HT, TH, and TT. If the coins are fair, the probability of each outcome is 1/4. The the expected humber of heads is |
E(H) = 2⋅(1/4) + 1⋅(1/4) + 1⋅(1/4) + 0⋅(1/4) = 1. |
Another example: tossing three coins the possible outcomes are HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. If the coins are fair, the probability of each outcome is 1/8. The the expected humber of heads is |
E(H) = 3⋅(1/8) + 2⋅(1/8) + 2⋅(1/8) + 1⋅(1/8) + 2⋅(1/8) + 1⋅(1/8) + 1⋅(1/8) + 0⋅(1/8) = 3/2. |
So the expected value need not be a value that can occur. The expected value is the average of the outcomes, after many repetitions. |
For continuous distributions, such as the normal, the mean is more easily expressed as a moment, which in turn is expressed as an integral. |