Topic: Probability and Stochastic Processes
Customer arrivals at a bank follow a Poisson process with mean 10 per hour. What is the probability that time between two arrivals is greater than 10 minutes?
- A. $\exp(-10/6)\approx0.188$
- B. $\exp(-1)\approx0.368$
- C. $\exp(-5/6)\approx0.435$
- D. $1-\exp(-5/6)\approx0.565$
Answer: A. $\exp(-10/6)\approx0.188$
Explanation: For a Poisson process, the time between arrivals (inter-arrival time $T$) follows an exponential distribution with parameter $\lambda$.
Given arrival rate $\lambda = 10$ arrivals/hour. Converting to minutes, $\lambda = 10/60 = 1/6$ arrivals/minute.
The probability that the inter-arrival time is greater than $t$ minutes is $P(T > t) = e^{-\lambda t}$.
For $t = 10$ minutes, $P(T > 10) = e^{-(1/6) \times 10} = e^{-10/6}$.
$e^{-1.667} \approx 0.188$.