3. Let U1, U2,. be a sequence of independent Ber(p) random variables. Define Xo 0 and...
4. Let Z1, Z2,... be a sequence of independent standard normal random variables. De- fine Xo 0 and n=0, 1 , 2, . . . . TL: n+1 , The stochastic process Xn,n 0, 1,2,3 is a Markov chain, but with a continuous state space. (a) Find EXn and Var(X). (b) Give probability distribution of Xn (c) Find limn oo P(X, > є) for any e> 0. (d) Simulate two realisations of the Markov process from n = 0 until...
Let Z1, Z2, . . . be a sequence of independent standard normal random variables. Define X0 = 0 and Xn+1 = (nXn + (Zn+1))/ (n + 1) , n = 0, 1, 2, . . . . The stochastic process {Xn, n = 0, 1, 2, } is a Markov chain, but with a continuous state space. (a) Find E(Xn) and Var(Xn). (b) Give probability distribution of Xn. (c) Find limn→∞ P(Xn > epsilon) for any epsilon > 0.
(a) Suppose that Xi, X2,... are independent and identically distributed random variables each taking the value 1 with probability p and the value-1 with probability 1-p For n 1,2,..., define Yn -X1 + X2+ ...+Xn. Is {Yn) a Markov chain? If so, write down its state space and transition probability matrix. (b) Let Xı, X2, ues on [0,1,2,...) with probabilities pi-P(X5 Yn - min(X1, X2,.. .,Xn). Is {Yn) a Markov chain and transition probability matrix. be independent and identically distributed...
1 (10pts) Let U1, U2, ... ,Un be independent uniform random variables over [0, 0] with the probability density function (p.d.f). () = a 2 + [0, 03, 0 > 0. Let U(1), U(2), .-. ,U(n) be the order statistics. Also let X = U(1)/U(n) and Y = U(n)- (a) (5pts) Find the joint probability density function of (X, Y). (b) (5pts) From part (a), show that X and Y are independent variables.
3. If U1 and U2 are independent standard uniform random variables, show that the variables are independent and identically distributed from N(0, 1) (the standard normal distribution) [10 marks
Unif (0, 1) 5. Suppose U1 and U2 i= 1,2. Let X; = - log(1 - U;), i = 1,2. [0, 1], U are independent uniform random variables on (a) Show that X1 and X2 are independent exponential random variables with mean 1, X; ~ Еxp(1), і — 1,2. (b) Find the joint density function of Y1 = X1 + X2 and Y2 = X1/X2 and show that Y1 and Y2 are independent. Unif (0, 1) 5. Suppose U1 and...
Let Ui and U2 be independent random variables, each one distributed uniformly on Z be the minimum, Z = min{U1, U2} and W be the maximum, W = max{U1, U2}. Find the joint p.d.f of Z and W [0, 1]. Let Let Ui and U2 be independent random variables, each one distributed uniformly on Z be the minimum, Z = min{U1, U2} and W be the maximum, W = max{U1, U2}. Find the joint p.d.f of Z and W [0,...
Problem 7. Let U1,U2,... be independent random variables all uniformly distributed on the unit interval, and let N be the first integer n 2 2 such that Un > Un-1. Show that for each real number 0<u < 1 !-un . 1- e-". (a) P(Ui-u and N = n) = (b) PUI S u and N is even) Problem 7. Let U1,U2,... be independent random variables all uniformly distributed on the unit interval, and let N be the first integer...
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.
74. Let X1, X2, ... be a sequence of independent identically distributed contin- uous random variables. We say that a record occurs at time n if X > max(X1,..., Xn-1). That is, X, is a record if it is larger than each of X1, ... , Xn-1. Show (i) P{a record occurs at time n}=1/n; (ii) E[number of records by time n] = {}_1/i; (iii) Var(number of records by time n) = 2/_ (i - 1)/;2; (iv) Let N =...