N indepedent unit vectors of uniformly random directions are added in R^3. What is the expected squared length of the resulting vector?
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Since the mean of the unit vectors is $0$, the mean of the squares of the lengths of the vectors is the variance of the vectors: the variance is the mean of the squares minus the square of the mean.
The mean of the square of length of one vector is $1$ (it is always $1$). Since the vectors are independent, the variance of the sum is the sum of the variances, so the variance of the sum of $N$ unit vectors is $N$.
Thus, as did claims in his comment, the expected squared length of the sum of $N$ uniformly directed, random unit vectors is $N$.