Let f (x 1 ,x 2 ) be the density of a r vec X (X 1 ,X 2 ) We call a curve in a (x 1 ,x 2 ) plane a level curve if it is determined by an equation f (x 1 ,x 2 ) c,where c is a constant Argue that all points in this curve are equally likely to be values of X Show that for a bivariate normal distribut...

Since the density fx1x2 is constant on a level curve for any point x1x2 from this curve the probabil ...

Let f (x 1 ,x 2 ) be the density of a r.vec X = (X 1

Question:

Let f (x₁,x₂) be the density of a r.vec X = (X₁,X₂). We call a curve in a (x₁,x₂)-plane a level curve if it is determined by an equation f (x₁,x₂) = c, where c is a constant. Argue that all points in this curve are equally likely to be values of X. Show that for a bivariate normal distribution, the level curves are ellipses. Graph typical level curves for the following covariance matrices: