List of top Statistics Questions on Stochastic Processes asked in GATE ST

Let \(\{X_n:n\ge0\}\) be a homogeneous Markov chain with state space \(S=\{1,2,\ldots,7\}\) and transition probability matrix
\[ P= \begin{pmatrix} \frac{1}{3} & 0 & \frac{2}{3} & 0 & 0 & 0 & 0 \\ 0 & \frac{1}{3} & 0 & \frac{1}{3} & 0 & \frac{1}{3} & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 & 0 & 0 \\ 0 & \frac{1}{2} & 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 \\ \frac{1}{2} & 0 & 0 & 0 & \frac{1}{4} & \frac{1}{4} & 0 \\ 0 & \frac{2}{3} & 0 & \frac{1}{6} & 0 & \frac{1}{6} & 0 \\ \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 & 0 & \frac{1}{3} \end{pmatrix}. \]
Then which of the following statements is correct?

Let \(\{W(t):t\ge0\}\) be a standard Brownian motion, with \(W(0)=0\). Define
\[Z_1=W(1)+W(2)\quad\text{and}\quad Z_2=W(2)+W(3).\]
Let \(\rho\) be the correlation coefficient between \(Z_1\) and \(Z_2\). Then the value of \(10\rho\) is ______ (round off to two decimal places).

Let \(\{N(t);t\geq0\}\) be a homogeneous Poisson process with rate \(3\), and let \(T_1\) denote the first arrival time. Then which of the following statements is/are correct?