Random points inside a parallelogram

Question

I have a 4 side convex Polygon defined by 4 points in 2D, and I want to be able to generate random points inside it.

If it really simplifies the problem, I can limit

Answer 1

Do the points need to be uniformly distributed, or is any distribution ok?

Can the polygon be concave, or is it guarenteed to be convex?

If the answer to both the above is no, then pick any two of the vertexes and pick a random point on the line segment between them. This is limited to the line segements connecting the vertexes (ie, VERY non-uniform); you can do a bit better by picking a third vertex and then picking a point between that and the first point -- still non-uniform, but at least any point in the polygon is possible

Picking a random point on a line between two points is easy, just A + p(B-A), where A and B are the points and p is a random number between 0.0 and 1.0