P = | 0.5 0.3 0.2 | | 0.2 0.6 0.2 | | 0.1 0.4 0.5 |
3.2. Let X(t), t ≥ 0 be a stochastic process with X(t) = A cos(t) + B sin(t), where A and B are independent random variables with mean 0 and variance 1. Find E[X(t)] and Autocov(t, s).
Below are some sample solutions to exercises from the second edition of "Stochastic Processes" by Sheldon M. Ross: Sheldon M Ross Stochastic Process 2nd Edition Solution
E[X(t)] = E[A cos(t) + B sin(t)] = E[A] cos(t) + E[B] sin(t) = 0
4.3. Consider a Markov chain with states 0, 1, and 2, and transition probability matrix: P = | 0
P X0 = 0 = P^2 (0,2) = 0.5(0.2) + 0.3(0.2) + 0.2(0.5) = 0.1 + 0.06 + 0.1 = 0.26
Var(X) = E[X^2] - (E[X])^2 = ∫[0,1] x^2(2x) dx - (2/3)^2 = ∫[0,1] 2x^3 dx - 4/9 = (1/2)x^4 | [0,1] - 4/9 = 1/2 - 4/9 = 1/18 Below are some sample solutions to exercises from
Sheldon M. Ross's "Stochastic Processes" is a renowned textbook that provides an in-depth introduction to the field of stochastic processes. The second edition of this book is a comprehensive resource that covers a wide range of topics, including random variables, stochastic processes, Markov chains, and queueing theory.