5:50 Back HW2.pdf Q to keep a copy for yourself. 5. On the top left corner of each page, indicate your name, login ID and student number. 6. Staple all your pages. HOMEWORK ASSIGNMENT 2 1. (3 marks) This problem is a continuation of T3Q1 on 1-D Simple Random Walk. Refer to our course notation. (i) Show that Foo(s) = 1 - √√1-4pqs² for |s| < 1. (ii) Hence, or otherwise, compute fo for each p = (0, 1). (iii) For p 1/2, show that "fo = 8. 2. (5 marks) For k = 1, 2,...,, let S, denote the arrival time of the k-th event of a Poisson process, {X(t): t> 0}, with parameter A. Let g be an integrable function defined on [0, ∞). For t> 0, show that E 9(S g(u)du. 1≤k≤x(t) [Hint: Apply the Law of Iterated Expectation and Theorem 4.2.2.] 3. (5 marks) Let X(t) be a Yule process with positive birth parameter 3 that is observed at a random time U, where U is uniformly distributed over [0,3). Also assume that the Yule process X(t) and the random time U are independent, and X(0)=1. (i) Find P(X(U)=k) for k ≥ 1. (ii) Find E[X(U)]. 4. (7 marks) Assume that X(t) and Y(t) are two independent standard Brownian motion processes satisfying X(0) = 0 and Y(0) = 0. Let M(t) = maxo 2, X(t) ≤ x) = 1 − $ ( ²² −77), 2> 0, x < 2, where denotes the cumulative distribution function of a standard normal. (ii) Find the joint probability density function, fм(e),x(e)(, ), of M(t) and X(t) for t> 0. [Hint: Apply Reflection Principle for (i).] OOO Dashboard Calendar To-do Notifications Inbox 5:50 Back HW2.pdf Q to keep a copy for yourself. 5. On the top left corner of each page, indicate your name, login ID and student number. 6. Staple all your pages. HOMEWORK ASSIGNMENT 2 1. (3 marks) This problem is a continuation of T3Q1 on 1-D Simple Random Walk. Refer to our course notation. (i) Show that Foo(s) = 1 - √√1-4pqs² for |s| < 1. (ii) Hence, or otherwise, compute fo for each p = (0, 1). (iii) For p 1/2, show that "fo = 8. 2. (5 marks) For k = 1, 2,...,, let S, denote the arrival time of the k-th event of a Poisson process, {X(t): t> 0}, with parameter A. Let g be an integrable function defined on [0, ∞). For t> 0, show that E 9(S g(u)du. 1≤k≤x(t) [Hint: Apply the Law of Iterated Expectation and Theorem 4.2.2.] 3. (5 marks) Let X(t) be a Yule process with positive birth parameter 3 that is observed at a random time U, where U is uniformly distributed over [0,3). Also assume that the Yule process X(t) and the random time U are independent, and X(0)=1. (i) Find P(X(U)=k) for k ≥ 1. (ii) Find E[X(U)]. 4. (7 marks) Assume that X(t) and Y(t) are two independent standard Brownian motion processes satisfying X(0) = 0 and Y(0) = 0. Let M(t) = maxo 2, X(t) ≤ x) = 1 − $ ( ²² −77), 2> 0, x < 2, where denotes the cumulative distribution function of a standard normal. (ii) Find the joint probability density function, fм(e),x(e)(, ), of M(t) and X(t) for t> 0. [Hint: Apply Reflection Principle for (i).] OOO Dashboard Calendar To-do Notifications Inbox