4. Let (Xi(t), t > 0) and (X2(t),t > 0) be two independent continuous Markov chains...
PROB 4 Let Xi and X2 be independent exponential random variables each having parameter 1 i.e. fx(x) = le-21, x > 0, (i = 1,2). Let Y1 = X1 + X2 and Y2 = ex. Find the joint p.d.f of Yi and Y2.
Let T: P2 --> R2 be the linear transformation such that T(x+1)=(1,1), T(x2)=(1,0) and T(x-1)=(0, 1). Find T(2+x+x2).
Define a Markov Chain on S = {0, 1, 2, 3, . . .} with transition probabilities p0,1 = 1, pi,i+1 = 1 − pi,i−1 = p, i ≥ 1 with 0 < p < 1. (a) Is the MC irreducible? (b) For which values of p the Markov Chain is reversible? 6. Define a Markov Chain on S 0, 1,2, 3,...) with transition probabilities i>1 with 0<p<. (a) Is the MC irreducible? (b) For which values of p the...
Consider a three-state continuous-time Markov chain in which the transition rates are given by The states are labelled 1, 2 and 3. (a) Write down the transition matrix of the corresponding embedded Markov chain as well as the transition rates out of each of the three states. (b) Use the symmetry of Q to argue that this setting can be reduced to one with only 2 states. (c) Use the results of Problem 1 to solve the backward equations of...
Consider a three-state continuous-time Markov chain in which the transition rates are given by The states are labelled 1, 2 and 3. (a) Write down the transition matrix of the corresponding embedded Markov chain as well as the transition rates out of each of the three states. (b) Use the symmetry of Q to argue that this setting can be reduced to one with only 2 states. (c) Use the results of Problem 1 to solve the backward equations of...
Consider a two state Markov chain with one-step transition matrix on the states 1,21, , 0<p+q<2. 91-9 ' Show, by induction or otherwise, that the n-step transition matrix is Ptg -99 Based upon the above equation, what is lim-x P(Xn-2K-1). How about limn→x P(Xn-
Let Xi and X2 be two continuous random variables having the joint probability density f,2)10 0, elsewhere. a. the joint pdf o1% and Y2.9(Y1,Y2), b, the P06 > Yi), c. the marginal pdfs gn () and g2(2), d. the conditional pdf h(walvi), and e. the E(Yalki-y) and E(gYi = 1/2).
(a) Is the MC irreducible? (b) For which values of p the Markov Chain is reversible? 6. Define a Markov Chain on S-10,1,2,3,...) with transition probabilities Po,1 pi,i+1 1 -pi,i-1 = p, i i>1 1 = with 0<p<
1. Let Xi, X2,... be independent random variables each with the standard normal distribution, and for each n 2 0 let Sn-1 Xi. Use importance sampling to obtain good estimates for each of the following probabilities: (a) Pfmaxn<100 Sn> 10; and (b) Pímaxns100 Sn > 30) HINTS: The basic identity of importance sampling implies that d.P n100 where Po is the probability measure under which the random variables Xi, X2,... are independent normals with mean 0 amd variance 1. The...
Let Xo, X1,... be a Markov chain with transition matrix 1(0 1 0 P 2 0 0 1 for 0< p< 1. Let g be a function defined by g(x) =亻1, if x = 1, if x = 2.3. , Let Yn = g(x,), for n 0. Show that Yo, Xi, is not a Markov chain.