List of top Probability Questions on Poisson Process

Consider an amusement park where visitors are arriving according to a Poisson process with rate 1. Upon arrival, a visitor spends a random amount of time in the park and then departs. The time spent by the visitors are independent of one another, as well as of the arrival process, and have common probability density function \[ f(x) = \begin{cases} e^{-x}, & x>0,
0, & \text{otherwise}. \end{cases} \] If at a given time point, there are 10 visitors in the park and \( p \) is the probability that there will be exactly two more visitors before the next departure, then \[ p = _________ \text{ (round off to 2 decimal places).} \]

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?