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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2009 • Question 4

Question

Math 6010 • Linear Models Spring 2009 • Question 4

Let \(X_1,X_2,X_3,Y_1,Y_2,Y_3\) be independent \(N(0,1)\) random variables. Let \(P_1\), \(P_2\), and \(P_3\) be the points in the plane with coordinates \((X_1,Y_1)\), \((X_2,Y_2)\), and \((X_3,Y_3)\). Let \(M\) be the midpoint of the segment \(P_1P_2\) and let \(r\) be half its length. Show that the probability that the point \(P_3\) lies within the circle of radius \(r\) centered at \(M\) is equal to \(1/4\).

Raw LaTeX

Let \(X_1,X_2,X_3,Y_1,Y_2,Y_3\) be independent \(N(0,1)\) random
variables. Let \(P_1\), \(P_2\), and \(P_3\) be the points in the plane with
coordinates \((X_1,Y_1)\), \((X_2,Y_2)\), and \((X_3,Y_3)\). Let \(M\) be the midpoint
of the segment \(P_1P_2\) and let \(r\) be half its length. Show that the
probability that the point \(P_3\) lies within the circle of radius \(r\) centered
at \(M\) is equal to \(1/4\).

Topic Tags

Classical Probability and Inclusion-Exclusion Covariance and Independence Measure-Theoretic Probability Uniform Integrability and Truncation Methods