Stochastic Methods I Homework 4 Due date: Friday September 29, 11:45 AM September 23, 2017 Your solutions must be submitted through Canvas as a single PDF document as an attachment in response to the homework assignment. Submit your work as a single high quality PDF document. 1. You are told that the probability space (, F, P) is of the following form: = 2 (two-dimensional Euclidean space), and the -field F consists of only 4 elements: F = {, 0, / A, Ac } , where A = {x 2 : x1 + x2 0}. You are also told that X is a random variable (nothing more). This implies that X is a function on , and a bit more (review your definitions!) In particular, X() cannot take on an infinite number of values as ranges over all possible values. (a) How many values can it take on? (b) Write an expression for the pmf for X, in terms of the symbols A and P. Note that strange -fields like this arise frequently in applications to \"state estimation\". This is an important theme in the course sequel. 2. Understanding the Poisson pmf. The rate of cars arriving at a tollbooth per minute is . A probabilistic model of this statement is the Poisson pmf: for any two consecutive times t, T (mins), the probability of k cars arriving in the interval (t, T ) is P(k;t, T ) = e k , k! where = (T t) . Important property: events in disjoint intervals are independent. Explanation: Consider the two disjoint intervals: (0,t1 ) and (t1 , T ), with 0 < t1 < T . Then for each pair of integers n1 , n2 , P[n1 cars in (0,t1 ) and n2 cars in (t1 , T )] = P[n1 cars in (0,t1 )]P[n2 cars in (t1 , T )] The proof of this statement is easy, after we learn a bit more about the origin of this pmf (be patient: you may assume independence holds in this exercise.) (a) Show that the following conditional probability does not depend on the value of : P[n1 cars in (0,t1 ) | n1 + n2 cars in (0, T )] (b) In (a) let T = 2, t1 = 1, n1 = n2 = 4. Compute P[4 cars in (0, 1) | 8 cars in (0, 2)] (c) Show that the probability that you found in part (a) is a binomial probability. In other words, show that the probability can be written as \u0012 \u0013 N n1 p (1 p)Nn1 . n1 Express N and p as functions of n1 , n2 ,t1 and t2 . (d) Give an intuitive explanation for the formula for p in part (c)