Question

Suppose that \( X \) has the probability density function \[ f(x) = \begin{cases} \frac{\lambda^{\alpha}}{\Gamma(\alpha)} x^{\alpha-1} e^{-\lambda x} & \text{if } x > 0, \\ 0 & \text{otherwise}, \end{cases} \] where \( \alpha > 0 \) and \( \lambda > 0 \). \text{Which one of the following statements is NOT true?}

• A. \( E(X) \) exists for all \( \alpha>0 \) and \( \lambda>0 \)
• B. Variance of \( X \) exists for all \( \alpha>0 \) and \( \lambda>0 \)
• C. \( E\left(\frac{1}{X}\right) \) exists for all \( \alpha>0 \) and \( \lambda>0 \)
• D. \( E(\log(1 + X)) \) exists for all \( \alpha>0 \) and \( \lambda>0 \)

Hint:

- The expectation of \( \frac{1}{X} \) does not always exist for all values of \( \alpha>0 \) and \( \lambda>0 \) due to the behavior of \( X \) near 0.
- Expectation functions involving logarithms or reciprocals can have existence conditions based on the tail behavior of the distribution.

Answer 1

The Correct Option is C

1) Understanding the probability density function:
This is the probability density function of a Gamma distribution with shape parameter \( \alpha \) and rate parameter \( \lambda \). The mean and variance of a Gamma-distributed random variable are well-defined for all \( \alpha>0 \) and \( \lambda>0 \).
2) Analysis of the options:
(A) \( E(X) \) exists for all \( \alpha>0 \) and \( \lambda>0 \):
This is true because the mean of a Gamma distribution exists for \( \alpha>0 \) and \( \lambda>0 \).
(B) Variance of \( X \) exists for all \( \alpha>0 \) and \( \lambda>0 \):
This is true because the variance of a Gamma distribution exists for \( \alpha>0 \) and \( \lambda>0 \).
(C) \( E\left(\frac{1}{X}\right) \) exists for all \( \alpha>0 \) and \( \lambda>0 \):
This is NOT true. For small values of \( X \), \( \frac{1}{X} \) becomes very large, and the expectation \( E\left(\frac{1}{X}\right) \) does not exist for all values of \( \alpha>0 \) and \( \lambda>0 \).
(D) \( E(\log(1 + X)) \) exists for all \( \alpha>0 \) and \( \lambda>0 \):
This is true. The logarithmic transformation of a Gamma distribution's expectation exists for all \( \alpha>0 \) and \( \lambda>0 \).
The correct answer is (C).