Question

Let X₁,X₂ and X₃ be three independent and identically distributed random variables having U(0, 1) distribution. Then \(E[(\frac{\ln X_1}{\ln X_1 X_2 X_3})^2]\) equals

• A. \(\frac{1}{6}\)
• B. \(\frac{1}{3}\)
• C. \(\frac{1}{8}\)
• D. \(\frac{1}{4}\)

Answer 1

The Correct Option is A

To find \( E\left[\left(\frac{\ln X_1}{\ln X_1 X_2 X_3}\right)^2\right] \), where \( X_1, X_2, \text{ and } X_3 \) are independent and identically distributed (i.i.d.) random variables having a uniform distribution over the interval (0,1), we proceed as follows:

1. First, note that for a random variable \( X \sim U(0, 1) \), the probability density function (PDF) is \( f_X(x) = 1 \) for \( 0 < x < 1 \).
2. The key part of the problem is handling the expression within the expectation: \[ \ln X_1, \quad \ln (X_1 X_2 X_3) = \ln X_1 + \ln X_2 + \ln X_3. \]
3. Recognize that \( \ln X_i \) is a -1 exponential random variable because if \( X_i \sim U(0,1) \), then \( \ln X_i \sim \text{Exp}(1) \).
4. Now, using properties of exponential random variables, we know: \[ \text{If } \ln X_i \sim \text{Exp}(1), \text{ then } \ln X_1 + \ln X_2 + \ln X_3 \sim \text{Gamma}(3, 1). \]
5. For independent random variables having the same rate parameter, their sum is a gamma distribution with scale parameter (inverse of rate parameter) the same (here 1) and shape parameter equal to the number of random variables (here 3).
6. Now, consider: \[ Z = \frac{\ln X_1}{\ln (X_1 X_2 X_3)} = \frac{\ln X_1}{\ln X_1 + \ln X_2 + \ln X_3}. \] Since all \( \ln X_i \) have the same distribution, this simplifies to: \[ E\left[\left(\frac{\ln X_1}{\ln (X_1 X_2 X_3)}\right)^2\right] = E\left[\left(\frac{Y_1}{Y_1 + Y_2 + Y_3}\right)^2\right], \] where \( Y_i \sim \text{Exp}(1) \).
7. The expectation value \( E\left[\left(\frac{Y_1}{Y_1 + Y_2 + Y_3}\right)^2\right] \) can be derived using the symmetry property of gamma distributions and known results: \[ E\left[\left(\frac{Y_1}{Y_1 + Y_2 + Y_3}\right)^2\right] = \frac{1}{6}. \]
8. The correct answer is therefore \[ \frac{1}{6}. \]