Bivariate gaussian

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 end2. Contour of bivariate Gaussian. Sketch the contour defined by f(x, y) = 0.06, where f(x, y) is the bivariate normal density with (a) Hr = 1, My = 2,02 = 03 = 1 and Pry = 0. (b) Hz = 0, My = 0, 02 = 02 = 1 and Pry = 0.2. (c) Hx = 0, My = 0,02 = 03 = 1 and Pxy = 0.8. (d) Ax = 0, My = 0, 02 = 4, 03 = 1 and Pry = 0.8. Here, o2 := VarX, o2 := VarY, and the correlation coefficient is defined as Pry := E(X -HI) ( Y-Hy) What is the distribution of X - Y, when (X, Y) has each of the bivariate distributions given above?7. (12 points) Consider a birth and death process with birth rate i + 1 and death rate i when there are i individuals. (a) Starting at state 2, find the expected time until the process first leaves state 2. (b) Find the expected time to go from state 2 to state 4.Exercises 4.1. In a birth and death process with birth parameters A,, = A for n = 0, 1, . . . and death parameters /,, = un for n = 0, 1, . . ., we have Po, (t) = (Ap )'exp j! where p= -[1 - e-"g]. Verify that these transition probabilities satisfy the forward equations (4.4), with i = 0. Pho(t) = - doPi.o ( t) + M, Pil (1), (4.4) P'., (1) = >;-P.j-(1) - (1; + 1, ) P., (1 ) + Me;+ P.j + (1), jz1,Determine whether the following value is a continuous random variable, discrete random variable, or not a random variable. 4. The number of textbook authors now eating a meal b. The "yes" or "no" response to a survey question c. The height of a randomly selected person d. The number of light bulbs that burn out in the next year in a room with 20 bulbs e. The number of home runs in a baseball game f. The amount of rain in City A during July a. is the number of textbook authors now eating a meal a discrete random variable, continuous random variable, or not a random variable? O A. It is a continuous random variable O B. It is a discrete random variable. O C. It is not a random variable b. is the "you" or "no" response to a survey question a discrole random variable, continuous random variable, or not a random variable? A It is a discrete random variable. O.B, It is a continuous random variable, O C. It is not a random variable. c. is the height of a randomly selected person a discrete random variable, continuous random venable, or not a random variable? A. it is a discrete random variable. O B. It is a continuous random variable. C C, it has not a random variable d. le the number of light bulbs that burn out in the next year in a room with 20 bulbs a ciacrete random vanable, continuous random variable, or not a random vanable? O A. It is a disccle random variable