Now suppose that you play a game in which you roll 100 dice, and the payoff is the total number of dots showing on all the dice. What is the variance of this game?

Answer 1

We first find the variance on the roll on one die. Then, as the rolls are independent, the variance on 100 rolls is 100 times the variance on one roll.

There is a short-cut for finding the variance of discrete distributions:

∑ p_ix_i² - (∑ p_ix_i)² = 1/6(1² + 2² + 3² + 4² + 5² + 6²) - (1/6(1+2+3+4+5+6))² = 1/6(91) - (7/2)² = 91/6 - 49/4 = 35/12

Thus, the variance for the sum of 100 rolls is 100(35/12) = 875/3 = 291 2/3