Question

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 \[ C(5, 3) \text{(in integer) is equal to} ________. \]

Hint:

For a Poisson process, the covariance function is simply the minimum of the two time points.

Answer 1

Correct Answer: 12

For a Poisson process \( X(t) \), the covariance function \( C(t_1, t_2) \) is given by: \[ C(t_1, t_2) = \min(t_1, t_2). \] Thus, to calculate \( C(5, 3) \), we find: \[ C(5, 3) = \min(5, 3) = 3. \] Therefore, the value of \( C(5, 3) \) is \( \boxed{3} \).