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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?

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fair dice with $P(X=x)=\frac1n$? – Bhavish Suarez Nov 21 '12 at 15:37
surely it has to fair dice... – thkang Nov 21 '12 at 15:43
up vote 3 down vote accepted

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.

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