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(Fair six-sided die)
Find E[X] for a fair die roll.
Outcomes: 1, 2, 3, 4, 5, 6. Each has probability 1/6.
E[X] = 1×(1/6) + 2×(1/6) + 3×(1/6) + 4×(1/6) + 5×(1/6) + 6×(1/6)
E[X] = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 21/6 = 3.5.
You cannot roll 3.5, but over thousands of rolls, the average approaches 3.5.
