Density function for X^2Y^2 and max(X,Y)

[kingkong123]

Hi guys i am struggling in how to find the density function for X^2Y^2 and max(X,Y).
Here's the scenario:
Suppose a random variable X has the Uniform distribution on the interval [-1,1].
Suppose a random variable Y has the exponential distribution with parameter lamda=2.
X and Y are independent.

attempt to find pdf of x^2Y^2:
X has a pdf f(x)=1/2 (if -1<=x<=1), Y has a pdf g(y)=2e^(-2y) (if y>=0).
I then calculated the joint density of X and Y, h(x,y)=f(x)g(y)=e^(-2y) (if -1<=x<=1 and y>=0).
Now let Z=(XY)^2 then P(Z<=z)=P((XY)^2<=z)=P(-sqrt(z)<=XY<=sqrt(z)).
Then i don't know what to do. i don't know whether integrate my joint density function range from -sqrt(z) to sqrt(z) with respect to y. After that i will then try to differentiate it to find density of (XY)^2

I don't know where to start to find density of max(X,Y).
Any help would be grateful. thx

 

[Stephen Tashi]

Do you remember from calculus how to solve problems like: Find the volume under the surface z = x^2 y^2 that lies over the circle bounded by x^2 + y^2 = 3 ? That would be done as a double integral. To do your problem that way is somewhat tricky since you have other boundaries due to places where the joint density is 0, so you would integrate over half circles or half circles with their ends chopped of by x =-1 and x = 1. According to this approach, you would do a double integral of f(x)g(x) over such figures. Set up the limits of integration based on the surface x^2 y^2 and then integrate f(x)g(x) over those limits.

I don't know whether there is any way to do the problem by doing an integral with respect to z and then taking partial derivatives or something like that.