Question

Suppose \( X_1, X_2, \dots, X_n, \dots \) are independent exponential random variables with the mean \( \frac{1}{2} \). Let the notation \( i.o. \) denote "infinitely often." Then which of the following is/are true?

• A. \( P \left( \left\{ X_n>\frac{\epsilon}{2} \log n \right\} \, i.o. \right) = 1 \text{ for } 0<\epsilon \leq 1 \)
• B. \( P \left( \left\{ X_n<\frac{\epsilon}{2} \log n \right\} \, i.o. \right) = 1 \text{ for } 0<\epsilon \leq 1 \)
• C. \( P \left( \left\{ X_n>\frac{\epsilon}{2} \log n \right\} \, i.o. \right) = 1 \text{ for } \epsilon>1 \)
• D. \( P \left( \left\{ X_n<\frac{\epsilon}{2} \log n \right\} \, i.o. \right) = 1 \text{ for } \epsilon>1 \)

Hint:

When working with exponential random variables, remember that the probability of an event occurring decreases exponentially as the value increases. This property is key to analyzing such problems involving infinite occurrences.

Answer 1

The Correct Option is A, B, D

Step 1: Understanding the Problem.
The problem deals with independent exponential random variables \( X_n \) with the mean \( \frac{1}{2} \). We need to find the probability that the event \( X_n \) satisfies certain conditions infinitely often. The notation \( i.o. \) means the event happens infinitely often as \( n \to \infty \).
Step 2: The Exponential Distribution.
For an exponential random variable with mean \( \lambda = \frac{1}{2} \), the probability density function (PDF) is given by: \[ f(x) = 2e^{-2x}, \quad x \geq 0 \] Thus, the cumulative distribution function (CDF) is: \[ P(X_n \leq x) = 1 - e^{-2x} \] From this, we can calculate the probabilities for different conditions.
Step 3: Analyze the Options.
- (A) \( P \left( \left\{ X_n>\frac{\epsilon}{2} \log n \right\} \, i.o. \right) = 1 \text{ for } 0<\epsilon \leq 1 \): This is true because the probability \( P(X_n>x) \) for an exponential random variable decreases exponentially, and for values of \( \epsilon \leq 1 \), the event will happen infinitely often with probability 1. - (B) \( P \left( \left\{ X_n<\frac{\epsilon}{2} \log n \right\} \, i.o. \right) = 1 \text{ for } 0<\epsilon \leq 1 \): This is also true because the probability \( P(X_n<x) \) for small \( x \) will be large enough to ensure the event happens infinitely often. - (C) \( P \left( \left\{ X_n>\frac{\epsilon}{2} \log n \right\} \, i.o. \right) = 1 \text{ for } \epsilon>1 \): This is false because for larger values of \( \epsilon \), the probability decreases significantly, and it is not guaranteed that the event will happen infinitely often. - (D) \( P \left( \left\{ X_n<\frac{\epsilon}{2} \log n \right\} \, i.o. \right) = 1 \text{ for } \epsilon>1 \): This is true because the probability of \( X_n \) being smaller than \( \frac{\epsilon}{2} \log n \) is high for larger \( \epsilon \), ensuring the event happens infinitely often.
Step 4: Conclusion.
Therefore, the correct options are (A), (B), and (D).