Question

Use Multivariate Transformations 7. (10 marks) The joint density of X and Y is given as f(x,y)=10, else a) (5 marks) Find the joint probability density function of = X / y and V = X. b) (5 marks) Using your result in (a), find the marginal density of V.

Answer 1

We will use Jacobian method to find the joint probability density function of U and V. It is given as

$f(u, v) = f(x,y) |J|$

where,

x, y are the solutions of the equation u = x/y and v = xy

and

$J = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}$

For 0 < y < 1 and 0 < x < 1

u can range from 0 to $\infty$ and v can range from 0 to 1