Sum of Dice

A dice is a discrete uniform from {1, 2, …, 6}

For X ~ U(a, a+1, …, b),

\[ \mathbb{E}[X] = \frac{a+b}{2}. \]

\[ \text{Var}(X) = \frac{(b-a+1)^2 - 1}{12}. \]

Clearly for a dice,

Var(X) = 35/12

For the sum of n IID X ~ U(a, a+1, …, b), by CLT

\[ S \sim N\left(n \cdot \frac{a+b}{2}, n \cdot \frac{(b-a+1)^2 - 1}{12}\right). \]

Applicable on fair coin, where a = 0, b = 1,

\[ S \sim N\left(\frac{n}{2}, \frac{n}{4}\right). \]

Applicable on fair dice, where a = 1, b = 6,

\[ S \sim N\left(3.5 \cdot n, \frac{35}{12} \cdot n\right). \]

Sum of N dice, or N times one dice, which is bigger?

Same expectation

Var(S1) = N^2 * 35/12 = N Var(S2)