Question

Let the random variable \( X \) have uniform distribution on the interval \( (0, 1) \) and \( Y = -2 \log X \). Then \( E(Y) \) equals

Hint:

For a uniform distribution on \( (0, 1) \), the expected value of \( \log X \) is -1, which simplifies the computation for transformations of this form.

Answer 1

Correct Answer: 2

Step 1: Understanding the uniform distribution.
The probability density function (PDF) of a uniform random variable \( X \) on \( (0, 1) \) is given by: \[ f_X(x) = 1 \quad \text{for} \quad 0<x<1. \]
Step 2: Define the transformation.
The random variable \( Y = -2 \log X \). To find \( E(Y) \), we compute the expected value of \( Y \), which is: \[ E(Y) = E(-2 \log X) = -2 E(\log X). \]
Step 3: Find \( E(\log X) \).
Since \( X \) is uniformly distributed on \( (0, 1) \), we have: \[ E(\log X) = \int_0^1 \log(x) \, dx = -1. \]
Step 4: Conclusion.
Thus, \( E(Y) = -2 \times (-1) = 2 \).