from my calculation if 4 6-sided dice are thrown:
$E[x] = 14.0 $ $E[x^2] = 207.66666.. $ $Var(x) = E[x^2]-(E[x])^2 = 11.666666666666657$
would there be some general formula for a dice with each dice having n sides?
from my calculation if 4 6-sided dice are thrown:
$E[x] = 14.0 $ $E[x^2] = 207.66666.. $ $Var(x) = E[x^2]-(E[x])^2 = 11.666666666666657$
would there be some general formula for a dice with each dice having n sides?
For a fair die with $n$ sides, we get $ E[x]=\sum_{i=1}^n \frac{1}{n} i =\frac{1}{n}\sum_{i=1}^n i=\frac{1+n}{2}$ And $ E[x^2]=\sum_{i=1}^n \frac{1}{n} i^2 =\frac{1}{n}\sum_{i=1}^n i^2=\frac{1}{6} (1+n) (1+2 n)$
And of course $Var[x]= \frac{1}{6} (1+n) (1+2 n)- (\frac{1+n}{2})^2 $
By the way, since the dice are independent, $Var[X_1 + X_2 + X_3 + X_4 ] = 4 Var[X_1]$ no matter how many sides the dice have. So you only need to calculate for one die and multiply.