Finding Density Functions for Randomly Chosen Points in a Unit Square

[Fuquan22]

Homework Statement

A point Q is chosen at random inside the unit square. What is the density function of the sum of the coordinates of point Q? What is the density function of the product of the coordinates of the point Q? Use geometry to find these densities.

Homework Equations

P(a < X < b) = Integral (a,b) of f(x)dx

The Attempt at a Solution

I know that the interval for the first part has to be between 0 and 2, but I don't know f(x)

 

[lanedance]

choosing a point at random from unit square is in effect equivalent to choosing 2 independent uniform random variables, say X & Y, on the interval [0,1].

Then you want to find the density function for Z = X + Y

 

[Fuquan22]

Yes but how do i go about finding that density function? I just don't know the final step to calculating a density function.

 

[zachzach]

It says use geometry. So does drawing lines of constant x+y help you solve it?

 

[Legendre]

I asked the same question in this thread:

https://www.physicsforums.com/showthread.php?t=398186

Hope it helps!