Problem 0.1 Let Xt be the number of people who enter a bank by time t > 0. Suppose Pr[Xt = k] ... Problem 0.1 Let Xt be the number of people who enter a bank by time t > 0. Suppose Pr[Xt = k] = (tk e−t )/k! , for k = [0, 1, 2, . . . ,] and Pr[Xt = k, Xs = r] = sr *(t − s)k−r *e−t /(r!(k − r)!) , for t > s > 0, and k ≥ r = 0, 1, 2, . . .
(a) Find E[X2 | X1 = 1].
Problem 0.1 Let Xt be the number of people who enter a bank by time t...
Problem 0.1 Let Xt be the number of people who enter a bank by time t > 0. Suppose Pr[Xt = k] = (t k e −t )/k! , for k = [0, 1, 2, . . . ,] and Pr[Xt = k, Xs = r] = sr *(t − s)k−r *e−t /(r!(k − r)!) , for t > s > 0, and k ≥ r = 0, 1, 2, . . . . (a) Find Pr[X2 = k |...
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!
Problem 0.1 Let X be the number of people who enter a bank by time t>0. Suppose ke-t k! for k 0,1,2,., and for t>s > 0, and k-r=0,1,2, . . . . (a) Find Pr(X2 = k | X,-1) for k = 0, 1, 2, . . . . (b) Find E[X2 X1-1 Useful information: Don't eat yellow snow, and et-L=0 tk/k! Problem 0.2 Recall the Geometric(p) distribution where Xnumber of flips of a coin until you get a...
Problem o.1 Let X, be the number of people who enter a bank by time t > 0. Suppose k! for k- 0,1,2,..., and s (t - s)k-e-t for t>s> 0, and k2r 0,1,2,.... (a) Find Pr[X2 k| X 1 for k 0,1,2,.... (b) Find E2 X1 1 Useful information: Don't eat yellow snow, andeot/k! Problem o.2 Recall the Geometric(p) distribution where X- number of flips of a coin until you get a head (H) with Pr(H) - p. The...
Let P(t) represent the number of people who, at time t, are infected with a certain disease. Let N denote the total number of people in the population. Assume that the spread of the disease can be modeled by the initial value problem: dP/dt = k(N − P)P, P(0) = P0. At time t = 0, when 100,000 member of a population of 500,000 are known to be infected, medical authorities intervene with medical treatment. As a consequence of this...
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
Question 7 (Chapters 6-7) 2+2+2+3+2+4+4-19 mark Let 0メs c Rn and fix r' E S. For a R" consider the following optimization problem: (Pa) min ar res and define the set K(S,) (aER z" is a solution of (Pa)) (e) If z' e int(S), prove that K(S, (0) (1) If possible, find a set S CR" and s* E S such that K(S,) (g) Let SB, 0.1] (rR l2l3 1) (the closed (, unit ball) and consider (1,0)7. Prove that...
Let x be an arithmetic brownian motion starting from 0 with
drift parameter 0.2
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
The sample data x1,x2,...,xn sometimes represents a time series, where xt = the observed value of a response variable x at time t. Often the observed series shows a great deal of random variation, which makes it difficult to study longer-term behavior. In such situations, it is desirable to produce a smoothed version of the series. One technique for doing so involves exponential smoothing. The value of a smoothing constant α is chosen (0 < α < 1). Then with...
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...