What is the average and variation of $20$ dices?

Question

If I roll a dice the average is

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

and

$$E(X^2) = (1+4+9+16+25+36)/6 = 91/6$$

$$VAR(x) = E(X^2) - (E(X))^2 = 91/6 - 49/4 = 35/12$$

Now the question is: How I can find the average and variation of $20$ dice rolls?

Answer 1

Let X1,X2,...X20 be the result of your 20 rolls.

You want to calculate $E(\sum_{i=1}^{20}{X_i})$ and $Var(\sum_{i=1}^{20}{X_i})$

$E(\sum_{i=1}^{20}{X_i})=\sum _{i=1}^{20}E(X_i)=20\times \frac{7}{2}=70$

$Var(\sum_{i=1}^{20}{X_i})=\sum _{i=1}^{20}Var(X_i)=20\times \frac{35}{12}=\frac{175}{3}$