Introduction to Probability · Problem 23

Sums of Random Variables — Grinstead & Snell

Mathematics — Probability · bivariate-normal-construction

Problem

*Let XX and VV be two standard normal random variables. Let ρ\rho be a real number between −1-1 and 11. (a) Let Y=ρX+1−ρ2 VY = \rho X + \sqrt{1 - \rho^2}\, V. Show that E(Y)=0E(Y) = 0 and Var(Y)=1Var(Y) = 1. We shall see later (see Example 7.5 and Example 10.17), that the sum of two independent normal random variables is again normal. Thus, assuming this fact, we have shown that YY is standard normal. (b) Using Exercises 17 and 18, show that the correlation of XX and YY is ρ\rho. (c) In Exercise 20, the joint density function fX,Y(x,y)f_{X,Y}(x, y) for the random variable (X,Y)(X, Y) is given. Now suppose that we want to know the set of points (x,y)(x, y) in the xyxy-plane such that fX,Y(x,y)=Cf_{X,Y}(x, y) = C for some constant CC. 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 (X,Y)(X, Y) are equally likely to fall. Show that for a given CC, the set of points of constant density is a curve whose equation is x2−2ρxy+y2=D ,x^2 - 2\rho xy + y^2 = D\,, where DD is a constant which depends upon CC. (This curve is an ellipse.) (d) One can plot the ellipse in part (c) by using the parametric equations x=rcos⁡θ2(1−ρ)+rsin⁡θ2(1+ρ) ,x = \frac{r\cos\theta}{\sqrt{2(1-\rho)}} + \frac{r\sin\theta}{\sqrt{2(1+\rho)}}\,, y=rcos⁡θ2(1−ρ)−rsin⁡θ2(1+ρ) .y = \frac{r\cos\theta}{\sqrt{2(1-\rho)}} - \frac{r\sin\theta}{\sqrt{2(1+\rho)}}\,. Write a program to plot 1000 pairs (X,Y)(X, Y) for ρ=−1/2,0,1/2\rho = -1/2, 0, 1/2. For each plot, have your program plot the above parametric curves for r=1,2,3r = 1, 2, 3.

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