DO NOT COPY OTHER ANSEWERS!!!! 2. (10 points) Let (%)n>o be a simple symmetric random walk....
4. Let {Sn,n > 0} be a symmetric Random Walk on Z. with So-0. Defined Y, max{Sk, 1 3 k S nt, for n 2 0, prove, thanks to a counterexample, that Y is not a Markov Chain
4. Let {Sn, n > 0} be a symmetric Random Walk on Z. with So-0. Defined Y max{Sk, 1 Sk n, for n 2 0, prove, thanks to a counterexample, that Y is not a Markov Chain.
3. Let(Sn, n > 0} be a symmetric Random Walk on Z. Defined To-inf(n-1 : Sn-0) the time of first passage to state 0, prove that PlT, = 2nlSo = 0] = 2n.plsøn = 이So = 0] for any n 2 1
3. Let {Sn, n > 0} be a symmetric Random Walk on Z. Defined To inf(n > 1 : Sn-0} the time of first passage to state 0, prove that 2n - 1 for any n 2 1
Problem 5.2 (10 points) For the simple symmetric random walk (Sn)n=0.12 that with So = 0, show for all n>0 and all -n<k<n Problem 5.2 (10 points) For the simple symmetric random walk (Sn)n=0.12 that with So = 0, show for all n>0 and all -n
Let us consider the biased random walk S,-X1 +X2+···+X,, with S: is a sequence of independent randon variables with P(X,--1)-g,P(X.-1) p+q=1 and Mt Show that M.-Sn-b- calculate Elr), where τ = inf{n : S" = a or-6) with a, b > 0. 0. where Xi, X2 p. where g)n is a martingale. Use this martingale to
(1) To test Ho: p=0.3; H :p > 0.3, a simple random sample of size n=200 is obtained from a population such that n < 0.05N. (a) If x = 75 and n=200, compute the test statistic zo. (b) Test the hypothesis using (i) the classical approach and (ii) the P-value approach. Assume an a= 0.05 level of significance. (c) What is the conclusion of the hypothesis test?
in 4. Suppose that {Xk, k > 1} is a sequence of i.i.d. random variables with P(X1 = +1) = 1. Let Sn = 2h=1 Xk (i.e. Sn, n > 1 is a symmetric simple random walk with steps Xk, k > 1). (a) Compute E[S+1|X1, ... , Xn] for n > 1. Hint: Check out Example 3.8 in the lecture notes (Version Mar/04/2020) for inspiration. (b) Find deterministic coefficients an, bn, Cn possibly depending on n so that Mn...
Let X be an exponential random variable such that P(X<26) = P(X > 26). Calculate E[X|X > 28]. Answer: CHECK
I. Let {X n\ be a sequence of random variables wit h E(X,-? for n- 7n exists a C > 0 such that for n 1,2, 3,.. Show that X is cons istent for ?