Examples of Stochastic Processes 2.3 2001/11/03 2002/07/09 00:00:00.003 GMT-5 Roy Ha rha@rice.edu Dinesh Rajan dinesh@ece.rice.edu Mohammad Jaber Borran mohammad@ece.rice.edu Behnaam Aazhang aaz@ece.rice.edu Roy Ha rha@rice.edu Dinesh Rajan dinesh@ece.rice.edu Mohammad Jaber Borran mohammad@ece.rice.edu Behnaam Aazhang aaz@ece.rice.edu stochastic process random process distribution stationary Examples of Stochastic processes. This modules contains examples of stochastic processes. For the material related to these examples, please refer to Introduction to Stochastic Processes. X t 2 f 0 t Θ ω where f 0 is the deterministic carrier frequency and Θ ω : ω is a random variable defined over and is assumed to be a uniform random variable; i.e., f 0 θ 1 2 0 F X t b X t b 2 f 0 t Θ b 2 f 0 t Θ b b 2 f 0 t Θ θ 2 f 0 t b 2 f 0 t 1 2 θ b 2 f 0 t 2 f 0 t 1 2 2 2 b 1 2 f X t x x 1 1 x 1 1 x 2 x 1 0 This process is stationary of order 1.

X t 2 f 0 t Θ X t 1 x 1 X t 2 b 2 ???? X t 1 x 1 2 f 0 t Θ Θ x 1 2 f 0 t X t 2 2 f 0 t 2 x 1 2 f 0 t 1 2 f 0 t 2 t 1 x 1

F X t 2 , X t 1 b 2 b 1 x 1 b 1 X t 1 x 1 X t 2 b 2 f X t 1 x 1 Every T seconds, a fair coin is tossed. If heads, then X t 1 for n T t n 1 T . If tails, then X t -1 for n T t n 1 T .

p X t x 1 2 x 1 1 2 x 1 X t is stationary of order 1. Second order probability mass function P X t 1 X t 2 x 1 x 2 p X t 2 | X t 2 x 2 | x 1 p X t 1 x 1 The conditional pmf p X t 2 | X t 1 x 2 | x 1 0 x 2 x 1 1 x 2 x 1 when n T t 1 n 1 T and n T t 2 n 1 T for some n. p X t 2 | X t 1 x 2 | x 1 p X t 2 x 2 for all x 1 and for all x 2 when n T t 1 n 1 T and m T t 2 m 1 T with n m P p X t 2 X t 1 x 2 x 1 0 x 2 x 1 n T t 1 t 2 n 1 T p X t 1 x 1 x 2 x 1 n T t 1 t 2 n 1 T p X t 2 x 2 p X t 1 x 1