Question

what is the expected value and standard deviation of a bet B that pays off the number of points on a fair die, squared? for example, if the die lands on 3, you receive $9.

Answer 1

Points on fair die	Payoff	Probability
1	1	1/6
2	4	1/6
3	9	1/6
4	16	1/6
5	25	1/6
6	36	1/6

Expected Payoff = E[P] = (1/6)*1 +(1/6)*4 + (1/6)*9 +(1/6)*16 + (1/6)*25 + (1/6)*36 = 91/6 = 15.16667

Variance of the payoff = (1/6)*(1-15.6667)² + (1/6)*(4-15.6667)² + (1/6)*(9-15.6667)² + (1/6)*(16-15.6667)² + (1/6)*(25-15.6667)² + (1/6)*(36-15.6667)² = 149.1389

Standard Deviation of the payoff is the squareroot of the variance

Standard deviation = 149.1389^1/2 = 12.21224

Answer (rounded to 2 decimals)

Expected Payoff = 15.17, Standard Deviation = 12.21