Question

Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent and identically distributed random variables each having uniform distribution on \( [0, 2] \). For \( n \geq 1 \), let \[ Z_n = - \log \left( \prod_{i=1}^{n} \left( 2 - X_i \right) \right)^{\frac{1}{n}}. \] Then, as \( n \to \infty \), the sequence \( \{Z_n\}_{n \geq 1} \) converges almost surely to _________ (round off to 2 decimal places).

Hint:

When dealing with products of random variables, consider applying the law of large numbers and expectation of logarithms for convergence.

Answer 1

Correct Answer: 0.27

As \( n \to \infty \), the sequence \( Z_n \) converges to the expectation of \( \log(2 - X_i) \), where \( X_i \) follows a uniform distribution on \( [0, 2] \). For a uniform distribution \( X \sim U(0, 2) \), the expected value is: \[ E[\log(2 - X)] = \int_0^2 \log(2 - x) \cdot \frac{1}{2} \, dx. \] This integral evaluates to approximately 0.27. Thus, \( Z_n \) converges almost surely to 0.27.