Read the fact patterns and answer the following questions.

i. Mr. and Mrs. Purple owned their matrimonial home as joint tenants. During their marriage, Mr. Purple demanded on several occasions that Mrs. Purple and her children (Mr. Purple's stepchildren) leave, only to later reconcile. Prior to his death, the husband obtained a divorce and applied for division of the matrimonial home according to relevant legislation, but Mr. Purple died before the property division occurred. The Administrator of the estate applied for property division under the relevant legislation. The issue that arose was whether or not the home formed part of Mr. Purple's estate or whether it passed to Mrs. Purple by way of survivorship.

a. Read Dunn Estate v. Dunn, along with the course text book and determine whether the house formed part of Mr. Purple's estate or it passed to Mrs. Purple by way of survivorship. Be sure to explain your answer, and provide examples where appropriate. (worth five marks)

ii. David Yellow owned and operated Yellow's General Hardware Store, as well as the land and buildings where he carried on business. His younger brother, John, worked for him as a manager. His daughter, Susan, also worked for Mr. Yellow. David died, leaving a will in which he gave his business and real property to John for life, with the remainder to Susan. John and Susan could not agree on how to run the business or take care of the property. Ultimately, John fired Susan. Over the next few years, the business did poorly and John allowed the warehouse and other buildings on the property to fall into a state of disrepair.

a. Susan commenced a lawsuit to force John to keep the buildings and property in a state of repair.What is the likelihood that she will be successful? Explain your answer. (worth five marks)

b. John died and in his will he left the entire estate, including the business and buildings to his wife. Who has entitlement in this situation? Explain your answer. (worth five marks)

4) You have learned in this course that one of the most practical applications of Derivatives is in the process of "Optimization Problems". You have been taught, in general, the key ideas, features, processes, and steps necessary to correctly solve an Optimization Problem. Given an optimization problem, explain how to go about solving it. In addition, explain under what circumstances your answers would be inadmissible/invalid based on the question being asked.Show that the following distribution is a member of the linear exponential family and identify each of the corresponding parameters (a(y), b(0), c(0), d(y) ) for the distribution. We refer to Eq.2 from the first page of this note Eq. 2 Zone parameter f ( y; 0) = explaly)b(0) + c(0) +d(y)] (Eq. 2) PROBLEM 1 (20POINTS) Suppose an experiment results in two possible outcomes (one being referred to as "success" and the other "failure"), with a success probability p. Show that the distribution is a member of the exponential family. The pmf is P (Y = y; p) = p(1-p)( 1-y) A) y y /2 a(v) = C ) b(p) = c(p) = d(y) = Vy D) y- 1 PROBLEM 2 (20POINTS) Suppose a counting process (with rate r) results in the pmf given below. Show that the distribution is a member of the exponential family. The pmf is 1 P ( Y = y;r) = ry y! -exp A) B ) a(v) = b(r) = c(r) = d(y) =Let A: Ill'OO MOII OMH a. Find the normalized eigenvectors and use them as columns in an orthogonal matrix C. b. Show that C'AC = D, where D is diagonal with the eigenvalues of A on the diagonal. c. Show that A = CDC'. 2. One of the core principles I want students to take away from the class is what we've called the "economic decision-making rule," and informs our theory of individual behavior. a. Write what the economic decision-making rule is and define marginal analysis. b. Define sunk costs and opportunity costs and discuss which of the two should be included when engaging in marginal analysis. c. Explain three contexts where we have seen the economic-decision make rule, interpret it in these contexts, and explain how they inform optimal decision-making (for example, a perfectly competitive firm sets P=MC to maximize profits, which is an application of the economic decision-making rule).Question 2 1. The Inverse Gaussian distribution is defined as f (I; M. 7) = V23XP Y(x - H )2 2u'r where u > 0 and y > 0. a) Show that the Inverse Gaussian distribution is a member of the exponential family. b) Give the natural parameter and the scale parameter. c) Hence, derive the mean and variance of Inverse Gaussian distri- bution. 2. Is the model log( E(Yi)) = Bo + Blog(I;). where Yi ~ IG(mi, y), for i e {1, 2,..., N} are independent inverse Gaussian distributed random variables, a generalized linear model? Give reasons for your