List of top Stochastic Processes Questions

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

In a laboratory experiment, cats choose between foods A and B. If a cat had A, it will prefer A next time with probability $0.7$; if it had B, it will prefer A next time with probability $0.5$. The experiment is repeated under identical conditions. If $40%$ cats had A in the first experiment, then the percentage (rounded off to one decimal place) of cats that will prefer A in the third experiment is __________.

A company sometimes stops payments of quarterly dividends. If the company pays the quarterly dividend, the probability that the next one will be paid is 0.7. If the company stops the quarterly dividend, the probability that the next quarterly dividend will not be paid is 0.5. Then the probability (rounded off to three decimal places) that the company will not pay quarterly dividend in the long run is ________

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 ________

Let \( \{ B(t) \}_{t \geq 0} \) be a standard Brownian motion and let \( \Phi(\cdot) \) be the cumulative distribution function of the standard normal distribution. If \[ P\left( \left( B(2) + 2B(3) \right)>1 \right) = 1 - \Phi\left( \frac{1}{\sqrt{\alpha}} \right), \, \alpha>0, \] then the value of \( \alpha \) (in integer) is equal to ________

Consider the following transition matrices \( P_1 \) and \( P_2 \) of two Markov chains: \[ P_1 = \begin{bmatrix} 1 & 0 & 0 \\ 1/3 & 1/2 & 1/6 \\ 0 & 0 & 1 \end{bmatrix} \quad \text{and} \quad P_2 = \begin{bmatrix} 1/6 & 1/3 & 1/2 \\ 1/4 & 0 & 3/4 \\ 0 & 1 & 0 \end{bmatrix}. \] Which of the following statements is correct?

Let \(\{X(t)\}_{t\geq 0}\) be a linear pure death process with death rate \(\mu_i = 5i\), \(i = 0, 1, \dots, N\), \(N \geq 1\). Suppose that \(p_i(t) = P(X(t) = i)\). Then the system of forward Kolmogorov’s equations is

Let \( W(t) \) be a standard Brownian motion. Then the variance of \( W(1)W(2) \) equals