Stochastic Processes 1. Let {Xt;t > 0} be a pure birth process with rate 1x >...
1. Let {Xt;t >0} be a pure birth process with rate 1x > 0, for x € S = {0,1,2,...}. (a) Write the backward equations (KBE) and use it to solve for Prz(t). (b) Use the result to part (a) to show that the waiting time in state x, say Wx, is exponentially distributed (c) Suppose 1x = 1 is constant for all x E S. Prove by induction that Px-kx(t) = (at) ke Af/k! for k = 0,..., and...
Stochastic processes problem Stochastic processes problem 1)Simulation of a Poisson process, let T,T2... be a succession of independent random variables with identical distribution exp() Define the random variable N of the following way: Prove that N have distribution Poisson(a)
Problem 0.1 Let Xt be the number of people who enter a bank by time> 0. Suppose t*e-! for k = 0,1,2 ,and for t > s > 0, and k > r = 0,1,2, (b) Find ElXyIX,-1]. Useful information: Don't eat yellow snow, and e- .*/k!
Let X-(Xt ,0 < t < 1} be an arithmetic Brownian motion starting from 0 with drift parameter μ-0.2 and variance parameter ơ2-0.125. 1. Calculate the probability that X2 is between 0.1 and 0.5 2. Given that X 0.6, find the probability that X2 is between 0.1 and 0.5 3. Given that Xi- 0.2, find the covariance between X2 and X3
(Stochastic process and probability theory) Let Xn, n > 1, denote a sequence of independent random variables with E(Xn) = p. Consider the sequence of random variables În = n(n-1) {x,x, which is an unbiased estimator of up. Does (a) in f H² ? (6) ûn 4* H?? (c) în + k in mean square? (d) Does the estimator în follow a normal distribution if n + ?
175-3. Consider the birth-and-death process with the following mean rates. The birth rates are Ao-2, A1 3,A.: 2. A 3 1, and A,s() for " > 3. The death rates are μ.-3,Pc-4. μ.-1,and = 2 for n > 4. (a) Construct the rate diagram for this birth-and-death process. (h) Develop the balance equations. (c) Solve these equations to find the steady-state probability dis- (di Use the general formulas for the birth-and-death process to cal- Also calculate L. L W.and
4. [20 points] Let {B(t):t0 be a standard Brownian motion. Define a stochastic process (X (t):t20) by the formulas X (t) = tB(1 + t-1)-tB(1), x(0) = 0, t > 0, You may take for granted the fact that imt-«HX(t) = 0, with probability 1 (b) Explain why [X():t20 is a standard Brownian motion 4. [20 points] Let {B(t):t0 be a standard Brownian motion. Define a stochastic process (X (t):t20) by the formulas X (t) = tB(1 + t-1)-tB(1), x(0)...
Consider a birth-and-death process with infinitesimal parameters Ae-5 for k> 0 and μk-15 for k 1. 1. Derive the limiting distribution of X(t), as t-oo 2. Find the limiting expectation of X(t), as t → oo. 3. Find the limiting variance of X (t)
1. Let {x, t,f 0) and {Yǐ.12 0) be independent Poisson processes,with rates λ and 2A, respectively. Obtain the conditionafdistributiono) Moreover, find EX Y X2t t given Yt-n, n = 1,2. 2, (a) Let T be an exponential random variable with parameter θ. For 12 0, compute (b) When Amelia walks from home to work, she has to cross the street at a certain point. Amelia needs a gap of a (units of time) in the traffic to cross the...
2. Heat equation Let ult, 2) satisfy the equation 4472(t, 2) +1, 0<r <1, t>0 with initial condition u(0,2) = 0, 0<x<1, and boundary conditions u(t,0) = 0, u(t,1)= 0, t> 0. This equation describes the temperature in a rod. The rod initially has a temperature of 0 (zero degree Celsius), and is then heated at a uniform rate 1. However, its two endpoints are kept at the temperature of 0 at all times. The unknown function u(t, x) describes...