[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Let [math]X[/math] and [math]V[/math] be two standard normal random variables. Let [math]\rho[/math]

be a real number between -1 and 1.

• Let [math]Y = \rho X + \sqrt{1 - \rho^2} V[/math]. Show that [math]E(Y) = 0[/math] and [math]Var(Y) = 1[/math]. We shall see later (see Example and Example), that the sum of two independent normal random variables is again normal. Thus, assuming this fact, we have shown that [math]Y[/math] is standard normal.
• Using Exercises Exercise and Exercise, show that the correlation of [math]X[/math] and [math]Y[/math] is [math]\rho[/math].
• In Exercise Exercise, the joint density function [math]f_{X,Y}(x, y)[/math] for the random variable [math](X, Y)[/math] is given. Now suppose that we want to know the set of points [math](x, y)[/math] in the [math]xy[/math]-plane such that [math]f_{X,Y}(x, y) = C[/math] for some constant [math]C[/math]. This set of points is called a set of constant density. Roughly speaking, a set of constant density is a set of points where the outcomes [math](X, Y)[/math] are equally likely to fall. Show that for a given [math]C[/math], the set of points of constant density is a curve whose equation is
  [[math]] x^2 - 2\rho x y + y^2 = D\ , [[/math]]
  where [math]D[/math] is a constant which depends upon [math]C[/math]. (This curve is an ellipse.)
• One can plot the ellipse in part (c) by using the parametric equations
  [[math]] \begin{eqnarray*} x & = & \frac {r\cos\theta}{\sqrt{2(1 - \rho)}} + \frac {r\sin\theta}{\sqrt{2(1 + \rho)}}\ , \\ y & = & \frac {r\cos\theta}{\sqrt{2(1 - \rho)}} - \frac {r\sin\theta}{\sqrt{2(1 + \rho)}}\ . \end{eqnarray*} [[/math]]
  Write a program to plot 1000 pairs [math](X, Y)[/math] for [math]\rho = -1/2, 0, 1/2[/math]. For each plot, have your program plot the above parametric curves for [math]r = 1, 2, 3[/math].