Residuals in Linear Regression and Assorted Exam-Style Questions

This section of sample problems and solutions is a part of The Actuary's Free Study Guide for Exam 3L, authored by Mr. Stolyarov. This is Section 61 of the Study Guide. See an index of all sections by following the link in this paragraph.

In least-squares linear regression, the ith residual is the difference between an observed value y_i when x = x_i and the value of y_i predicted by the least squares line. The greater the ith residual, the greater the failure of the least squares line to accurately model the location of the point

(x_i, y_i). The ith residuals for all values of i can be graphed to produce a residual plot.

Some of the problems in this section were designed to be similar to problems from past versions of the Casualty Actuarial Society's Exam 3L and the Society of Actuaries' Exam MLC. They use original exam questions as their inspiration - and the specific inspiration for each problem is cited so as to give students a chance to see the original. All of the original problems are publicly available, and students are encouraged to refer to them. But all of the values, names, conditions, and calculations in the problems here are the original work of Mr. Stolyarov.

Sources: Broverman, Sam. Actuarial Exam Solutions - CAS Exam 3L - Spring 2008.

Larsen, Richard J. and Morris L. Marx. An Introduction to Mathematical Statistics and Its Applications. Fourth Edition. Pearson Prentice Hall: 2006. pp. 650.

Original Problems and Solutions from The Actuary's Free Study Guide

Problem S3L61-1. Similar to Question 8 from the Casualty Actuarial Society's Spring 2008 Exam 3L. This question is also an excellent review of the concepts in Section 53.

You are taking a sample of 2 random values from an exponential distribution whose mean θ is equal to 700.

The ith order statistic of Y has the following probability density function (p. d. f.):
f_{Y'_i}(y) = (n!/((i -1)!(n - i)!))*F(y)^i-1*(1-F(y))^n-i*f(y), where f(y) is the p. d. f. of Y and F(y) is the cumulative distribution function (c. d. f.) of Y.

Find the mathematical expectation of the larger of the two values drawn from the sample.