Bata questions

Problem 1. Consider the problem [3 points] 1 + 13 =0 00 u(x, 0) = uo(X) 0 0 3 %Upwind method is used clear , clf , hold off %clear variables and graph N=100; %%number of gridpoints . hx=1/N; %stepsize in x-direction x x hx : hx :1; %-grid , inner points . xp =[0 x]: %x-grid plus boundary point sht =0.01; %timestep 10 mucht/hx; %Courant number 1 40 zeros (N. 1); %%IV at inner points 12 up = [1: 40 ]: %%IV+BV u plot (xp , up) u title( Initial state for the advection equation ) is xlabel ( 10 ylabel( u (x, t) 17 u (1 :N,1)=40; %initialization of upwind us for k=1:100 %take 100 timesteps with upwind Is u (1 , k+1)=(1-mu). u(1 , k)+mu=1;%-1 is .BC 20 u (2 :N, k+1)=(1-mu). u (2:N. k)+musu (1:N-1,k); a up =[1 :u (: . k+1)]: a plot (xp. up) %%plot the front for each timestep a hold on %keep the previous plots pause (0. 1) 25 end14. Consider the linear growth birth and death process X(t) with parameters 1, u and a = 0. Assume X(0) =1. Find the distribution of the number of living individuals at the time of the first death.2. (5pts) Describe a Poisson process as a birth and death process (specify the states and the rates). Define the corresponding instantaneous transition rates in a CTMC. Explain their meaning. Problem 2: 15 points = [5 + 5 + 5] Consider a birth-and-death process, X - {X(t) : t 2 0}, with instant rates Xx = A and uk = / per hour. You are given information about the expected departure times from states 0 and 1 as follows. . E [So |X(0) = 0] = 1 hour . E[Si |X(0) = 1] = 12 minutes 1. Determine parameters A and / 2. Derive the limiting distribution, A = lim P[X(() - *] for any * 2 0 3. Find the limiting expectation, lim E[X(t)] SolutionThe beta distribution with shape parameters a > 0 and f > 0 has pdf fx (x) = - B(a,B) for x E [0,1] where r(a)r(B) B(a. P) =T(a + 8) a) Show that the beta distribution exhibits mirror symmetry: if X -Beta(a, 8) then 1 - X-Beta (8, a) b) If X-Beta(a, 1) show that - In X ~Exp(a) c) If X~Beta(1,1) show that X-U(0,1) d) Show that the beta distribution Beta(a, () has mode XE 2-a-B