[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]

When studying certain physiological data, such as heights

of fathers and sons, it is often natural to assume that these data (e.g., the heights of the fathers and the heights of the sons) are described by random variables with normal densities. These random variables, however, are not independent but rather are correlated. For example, a two-dimensional standard normal density for correlated random variables has the form

[[math]] f_{X,Y}(x,y) = \frac 1{2\pi\sqrt{1 - \rho^2}} \cdot e^{-(x^2 - 2\rho xy + y^2)/2(1 - \rho^2)}\ . [[/math]]

• Show that [math]X[/math] and [math]Y[/math] each have standard normal densities.
• Show that the correlation of [math]X[/math] and [math]Y[/math] (see Exercise Exercise) is [math]\rho[/math].