Question

Let customers arrive at a departmental store according to a Poisson process with rate 10. Further, suppose that each arriving customer is either a male or a female with probability \( \frac{1}{2} \) each, independent of all other arrivals. Let \( N(t) \) denote the total number of customers who have arrived by time \( t \). Then which one of the following statements is NOT true?

• A. If \( S_2 \) denotes the time of arrival of the second female customer, then \( P(S_2 \leq 1) = 25 \int_0^1 se^{-5s} \, ds \)
• B. If \( M(t) \) denotes the number of male customers who have arrived by time \( t \), then \( P\left( M\left( \frac{1}{3} \right) = 0 \mid M(1) = 1 \right) = \frac{1}{3} \)
• C. \( E\left[ (N(t))^2 \right] = 100t^2 + 10t \)
• D. \( E[N(t)N(2t)] = 200t^2 + 10t \)

Hint:

In a Poisson process, conditional probabilities involving the number of arrivals in non-overlapping intervals can be computed using the memoryless property of the exponential distribution.

Answer 1

The Correct Option is B

- Option (A) is correct. \( S_2 \) is the time of the second female arrival in a Poisson process, and this follows the exponential distribution with rate \( 5 \), which leads to the correct probability expression.
- Option (B) is incorrect. Given that \( M(1) = 1 \), the probability that \( M\left( \frac{1}{3} \right) = 0 \) is not \( \frac{1}{3} \). The conditional probability is not that simple, and this statement is false.
- Option (C) is correct. The expectation \( E[(N(t))^2] \) for a Poisson process with rate \( 10 \) is given by \( E[(N(t))^2] = 100t^2 + 10t \).
- Option (D) is correct. The expected value \( E[N(t)N(2t)] \) for a Poisson process with rate \( 10 \) is \( 200t^2 + 10t \).
Final Answer: \[ \boxed{P\left( M\left( \frac{1}{3} \right) = 0 \mid M(1) = 1 \right) = \frac{1}{3}} \text{ is NOT true.} \]