# 2013-15 Bounded random variable

Let $$x, y$$ be real numbers satisfying $$y \geq x^2 + 1$$. Prove that there exists a bounded random variable $$Z$$ such that
$E[Z] = 0, E[Z^2] = 1, E[Z^3] = x, E[Z^4] = y.$
Here, $$E$$ denotes the expectation.

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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 $$N$$.