N fair 8-sided dice are independently cast. What is the expectation value for the sum of all the
top spots showing? What are the variance and standard deviation of the sum of all the top spots
showing?
Solution
For one roll, the average is (1+2+3+4+5+6+7+8)/8 = 36/8 = 9/2
For one roll, E(x^2) = (1+2^2+3^2+4^2+5^2+6^2+7^2+8^2)/8 = 51/2.
var(x) = E(x^2) - (E(x))^2 = 51/2 - (9/2)^2 = 51/2 - 81/4 = 21/4
The standard deviation is sqrt(var) = sqrt(21)/2
Then, if there are N fair die independently cast,
E(the sum of N die) = 9/2 N
variance sums with independence
var(the sum of N die) = 21/4 N
Then, standard deviation of the sum of N die = sqrt(21N/4) or sqrt(21N)/2.
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N fair 8-sided dice are independently cast. What is the expectation .pdf
1. N fair 8-sided dice are independently cast. What is the expectation value for the sum of all the
top spots showing? What are the variance and standard deviation of the sum of all the top spots
showing?
Solution
For one roll, the average is (1+2+3+4+5+6+7+8)/8 = 36/8 = 9/2
For one roll, E(x^2) = (1+2^2+3^2+4^2+5^2+6^2+7^2+8^2)/8 = 51/2.
var(x) = E(x^2) - (E(x))^2 = 51/2 - (9/2)^2 = 51/2 - 81/4 = 21/4
The standard deviation is sqrt(var) = sqrt(21)/2
Then, if there are N fair die independently cast,
E(the sum of N die) = 9/2 N
variance sums with independence
var(the sum of N die) = 21/4 N
Then, standard deviation of the sum of N die = sqrt(21N/4) or sqrt(21N)/2
