List of top Stochastic Processes Questions on Poisson Process

Consider a Poisson process \( \{ X(t), t \geq 0 \} \). The probability mass function of \( X(t) \) is given by \[ f(t) = \frac{e^{-4t} (4t)^n}{n!}, \quad n = 0, 1, 2, \dots \] If \( C(t_1, t_2) \) is the covariance function of the Poisson process, then the value of \[ C(5, 3) \text{(in integer) is equal to} ________. \]

At a telephone exchange, telephone calls arrive independently at an average rate of 1 call per minute, and the number of telephone calls follows a Poisson distribution. Five time intervals, each of duration 2 minutes, are chosen at random. Let \( p \) denote the probability that in each of the five time intervals at most 1 call arrives at the telephone exchange. Then \( e^{10} p \) (in integer) is equal to ________