Question

The following table shows the probability of rolling the numbers 1 through 7 on a fair seven-sided die.

X 1 2 3 4 5 6 7
P(X) 1/7 1/7 1/7 1/7 1/7 1/7 1/7
What is the standard deviation of the random variable X, "the number that comes up on the die?"

Answer 1

E(X) = 1/7 + 2(1/7) + 3(1/7) + 4(1/7) + 5(1/7) + 6(1/7) + 7(1/7) = 1/7 + 2/7 + 3/7 + 4/7 + 5/7 + 6/7 + 7/7 = 24/7 = 4.

Var(X) = 1/7(1 - 4)^2 + 1/7(2 - 4)^2 + 1/7(3 - 4)^2 + 1/7(4 - 4)^2 + 1/7(5 - 4)^2 + 1/7(6 - 4)^2 + 1/7(7 - 4)^2 = 9(1/7) + 4(1/7) + 1/7 + 1/7 + 4(1/7) + 9(1/7) = 9/7 + 4/7 + 2/7 + 4/7 + 9/7 = 28/7 = 4

standard deviation = sqrt(Var(X)) = sqrt(4) = 2.