Prove the following: There exists a bounded real random variable \( Z \) such that
\[
E[Z] = 0, E[Z^2] = 1, E[Z^3] = x, E[Z^4] = y
\]
if and only if \( y \geq x^2 + 1 \). (Here, \( E \) denotes the expectation.)
Tag Archives: random variable
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.
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 \( N \).