Question

Let \( X \) be a random variable with probability density function \[ f(x) = \begin{cases} e^{-x} & \text{if } x \geq 0 \\ 0 & \text{otherwise} \end{cases} \] For \( a < b \), if \( U(a, b) \) denotes the uniform distribution over the interval \( (a, b) \), then which of the following statements is/are true?

• A. \( e^{-X} \) follows \( U(-1, 0) \) distribution
• B. \( 1 - e^{-X} \) follows \( U(0, 2) \) distribution
• C. \( 2e^{-X} - 1 \) follows \( U(-1, 1) \) distribution
• D. The probability mass function of \( Y = [X] \) is \( P(Y = k) = (1 - e^{-1}) e^{-k} \text{ for } k = 0, 1, 2, .... \), where \( [X] \) denotes the largest integer not exceeding \( X \)

Hint:

- For transformations of random variables, remember to find the distribution of the transformed variable by using its cumulative distribution function (CDF) and differentiating it.
- The exponential distribution can be transformed into other distributions by simple algebraic manipulations, such as scaling and shifting.

Answer 1

The Correct Option is C

1) Understanding the problem:
The probability density function of \( X \) is that of an exponential distribution with rate 1. We are tasked with analyzing the transformations of \( X \) and determining the resulting distributions.

2) Analyzing each option:
- (A) \( e^{-X} \) follows \( U(-1, 0) \) distribution: This is false. The transformation \( e^{-X} \) will result in a distribution bounded between 0 and 1, not between -1 and 0.

- (B) \( 1 - e^{-X} \) follows \( U(0, 2) \) distribution: This is false. The transformation \( 1 - e^{-X} \) will result in a uniform distribution on \( (0, 1) \), not \( (0, 2) \).

- (C) \( 2e^{-X} - 1 \) follows \( U(-1, 1) \) distribution: This is true. The transformation \( 2e^{-X} - 1 \) maps the exponential distribution to a uniform distribution on the interval \( (-1, 1) \).

- (D) The probability mass function of \( Y = [X] \) is \( P(Y = k) = (1 - e^{-1}) e^{-k} \text{ for } k = 0, 1, 2, .... \): This is true. The floor function \( [X] \) gives the largest integer not exceeding \( X \), and the PMF follows the given form.

Thus, the correct answer is (C).