Question

Suppose that \( U \) and \( V \) are two independent and identically distributed random variables each having probability density function \[ f(x) = \begin{cases} \lambda^2 x e^{-\lambda x} & \text{if } x > 0 \\ 0 & \text{otherwise}, \end{cases} \] where \( \lambda > 0 \). Which of the following statements is/are true?

• A. The distribution of \( U - V \) is symmetric about 0
• B. The distribution of \( UV \) does not depend on \( \lambda \)
• C. The distribution of \( \frac{U}{V} \) does not depend on \( \lambda \)
• D. The distribution of \( \frac{U}{V} \) is symmetric about 1

Hint:

- The difference \( U - V \) of two i.i.d. exponential random variables is symmetric about 0.
- The ratio \( \frac{U}{V} \) of two i.i.d. exponential random variables follows a distribution that is independent of the rate parameter \( \lambda \).

Answer 1

The Correct Option is A

1) Understanding the problem:
The random variables \( U \) and \( V \) are independent and identically distributed (i.i.d.), each following an exponential distribution with parameter \( \lambda \). The given probability density function (PDF) is for the exponential distribution, which describes the waiting time between events in a Poisson process. The question asks us to identify the correct properties of the distribution of various combinations of \( U \) and \( V \).
2) Analysis of the options:
(A) Correct: The distribution of \( U - V \) is symmetric about 0.
Since \( U \) and \( V \) are i.i.d. random variables, the distribution of their difference \( U - V \) is symmetric about 0. This is because the exponential distribution is memoryless, and the difference of two i.i.d. random variables with identical distributions results in a symmetric distribution about zero.
(B) Incorrect: The distribution of \( UV \) depends on \( \lambda \).
The product of two exponential random variables will have a distribution that depends on \( \lambda \), as the rate parameter \( \lambda \) affects the scale of the distribution. Hence, the statement that the distribution of \( UV \) does not depend on \( \lambda \) is incorrect.
(C) Correct: The distribution of \( \frac{U}{V} \) does not depend on \( \lambda \).
The ratio of two independent exponential random variables follows a distribution that is independent of \( \lambda \), making this statement true. This is a well-known result for the ratio of two exponential random variables.
(D) Incorrect: The distribution of \( \frac{U}{V} \) is not symmetric about 1.
The ratio of two exponential random variables does not have symmetry about 1. It follows a different distribution that is not symmetric.
The correct answers are (A) and (C).