# 2013-10 Mean and variance of random variable

Let random variables $$\{ X_r : r \geq 1 \}$$ be independent and uniformly distributed on $$[0, 1]$$. Let $$0 < x < 1$$ and define a random variable $N = \min \{ n \geq 1 : X_1 + X_2 + \cdots + X_n > x \}.$
Find the mean and variance of $$N$$.

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Let $$n$$ be a fixed positive integer and let $$p\in (0,1)$$. Let $$D_n$$ be the determinant of a random $$n\times n$$ 0-1 matrix whose entries are independent identical random variables, each of which is 1 with the probability $$p$$ and 0 with the probability $$1-p$$.  Find the expected value and variance of $$D_n$$.