Question

Consider the following statements:
(I) Let a random variable \(X\) have the probability density function \[ f_X(x) = \frac{1}{2} e^{-|x|}, \quad -\infty<x<\infty. \] Then there exist i.i.d. random variables \(X_1\) and \(X_2\) such that \(X\) and \(X_1 - X_2\) have the same distribution.
(II) Let a random variable \(Y\) have the probability density function \[ f_Y(y) = \begin{cases} \frac{1}{4}, & -2<y<2, \\ 0, & \text{elsewhere.} \end{cases} \] Then there exist i.i.d. random variables \(Y_1\) and \(Y_2\) such that \(Y\) and \(Y_1 - Y_2\) have the same distribution.
Then which of the above statements is/are true?

• A. (I) only
• B. (II) only
• C. Both (I) and (II)
• D. Neither (I) nor (II)

Hint:

For the Laplace distribution, the difference of two independent variables with the same distribution follows the same distribution. However, for the uniform distribution, the difference follows a triangular distribution.

Answer 1

The Correct Option is A

We are given two statements about random variables \(X\) and \(Y\), and we need to determine which one is true.
Step 1: Analyze Statement (I)
The probability density function \(f_X(x) = \frac{1}{2} e^{-|x|}\) represents the Laplace distribution with mean 0 and variance 2. Let us now check if there exist i.i.d. random variables \(X_1\) and \(X_2\) such that \(X\) and \(X_1 - X_2\) have the same distribution. For Laplace-distributed random variables, it is known that the difference of two independent Laplace-distributed random variables with the same distribution is also Laplace-distributed. Thus, we can conclude that the difference \(X_1 - X_2\) has the same distribution as \(X\). Therefore, statement (I) is true.
Step 2: Analyze Statement (II)
The probability density function \(f_Y(y) = \frac{1}{4}\) for \( -2<y<2 \) represents a uniform distribution over the interval \([-2, 2]\). For the difference of two independent uniform random variables, \(Y_1 - Y_2\), the distribution is a triangular distribution, which does not have the same form as the uniform distribution. Thus, statement (II) is false.