Let's say I want to design a sampling procedure, which samples a real number from the sample space [0, X]. The sampling is done in 3 steps:
Step 1: Sample [0, X] M times using a given PDF, say PDF1.
Step 2: Sample [0, X] N times using another given PDF, say PDF2.
Step 3: Random sample the M+N samples to generate the final sample.
What's the overall PDF of my sampling strategy? It must be a function of PDF1, PDF2, M, and N, but I couldn't figure it out. Any thoughts or pointers?