This is an open book olen notes exam You can use a calculato

This is an open book, olen notes exam. You can use a calculator or computer, but you shouldn\'t need to. Good luck! 1. (20 points) suppose X has density f(r) 0 otherwise (a) Find c. do you generate X by the inverse transform method? (b) He 2. (20 points) write a Matlab program that estimates the expected d tance between two points chosen uniformly from the unit circle, 3. (0 points) Let P be a transition matrix for a Markov chain with states (1, 2,..., k). Suppose for each j u Pr 1. Prove that the station ary (steady state) probability vector is (t,t,..., t). 4. (50 points) Let (NI, N be i.i.d. random variables with P(N I) .5, P(Ni 2) 4 and P(N, 3) 1. Let Xo 0, and for n 0 let Xn (X, -1 Nin) mod 5, i.e., E (0, 1, 2, 3, 4) is the remainder after dividing by 5. (a) In a few words, explain why fxn) is a Markov chain (b) Write down the transition matrix for (xnh. (c) Find the stationary (steady state) probability vector. Hint: Use problem 3 (d) Write the equations to find the expected number of transitions it takes to get from state 0 back to state 0. (Do not solve.) (e) Write the equations to find the probability the process hits state 1 before returning to state 0, starting from state 0. (Do not solve.)

Solution

Given that f(x)= c x^2 when 0<=x<=1

=0 otherwise

Equating the both conditions c x^2 =0

substitute x =1

====> c*1 =0

====> c=0