1. Let K be a geometric random variable having probability mass function P(K = k)= (1-2)k-¹a, 0<^<1, k = 1,2,... (It is a discrete-time counter part of exponential random time with intensity 2). Consider a discrete-time stochastic process X = (X k= 1,2,...) where X, is defined by Xk (b) For 1 ≤ k < k₂ k₁ ≥ 1, P(K > K₂|K > K₁) = P(K > k₂-k₁). P(Xk3= 1|Xk₂ = 0, Xk, = 0) = P(Xk3 = 1|Xk₂ = 0), and conclude whether or not X, is a Markov process. (c) X is stationary in time, i.e., for 1 ≤ ki < k₂, P(Xk₂ = 1|Xk₁ = 0) = P(Xk2-k₁ = 1|Xo = 0). (d) Write the state space S of X and derive the (one-step) transition matrix.
1. Let K be a geometric random variable having probability mass function P(K = k)= (1-2)k-¹a, 0<^<1, k = 1,2,... (It is a discrete-time counter part of exponential random time with intensity 2). Consider a discrete-time stochastic process X = (X k= 1,2,...) where X, is defined by Xk (b) For 1 ≤ k < k₂ k₁ ≥ 1, P(K > K₂|K > K₁) = P(K > k₂-k₁). P(Xk3= 1|Xk₂ = 0, Xk, = 0) = P(Xk3 = 1|Xk₂ = 0), and conclude whether or not X, is a Markov process. (c) X is stationary in time, i.e., for 1 ≤ ki < k₂, P(Xk₂ = 1|Xk₁ = 0) = P(Xk2-k₁ = 1|Xo = 0). (d) Write the state space S of X and derive the (one-step) transition matrix.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:1. Let K be a geometric random variable having probability mass function
P(K = k) = (1 - 1)k-¹A, 0<< 1, k = 1,2,...
(It is a discrete-time counter part of exponential random time with intensity 1). Consider
a discrete-time stochastic process X = (X k = 1,2, ...) where X, is defined by
Xk =
(b) For 1 ≤k₁ <k₂ <k3,
1,
K≤k
0, otherwise
In short-hand notation, the process X is written as X = 1(K<k). Show that
(a) The random time K has memoryless property. Namely, for k₂ > k₁ ≥ 1,
P(K> k₂|K> k₁) = P(K>k₂-k₁).
P(Xk3= 1|Xk₂ = 0, Xk₁ = 0) = P(Xk3 = 1|Xk₂ = 0),
and conclude whether or not X is a Markov process.
(c) X is stationary in time, i.e., for 1 ≤ k₁ <k2,
P(Xkz = 1|Xk = 0) = P(Xz→k = 1|X = 0).
(d) Write the state space S of X and derive the (one-step) transition matrix.
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