As a generalization of Example 4.1, consider a test of n circuits such that each circuit is acceptable with probability p, independent of the outcome of any other test. Show that the joint PMF of X, the number of acceptable circuits, and Y, the number of acceptable circuits found before observing the first reject, is
Where
A: The first y tests are acceptable.
B: Test y + 1 is a rejection.
C: The remaining n - y - 1 tests yield x - y acceptable circuits
Example 4.1
Test two integrated circuits one after the other. On each test, the possible outcomes are a (accept) and r (reject). Assume that all circuits are acceptable with probability 0.9 and that the outcomes of successive tests are independent. Count the number of acceptable circuits X and count the number of successful tests Y before you observe the first reject. (If both tests are successful, let Y = 2.) Draw a tree diagram for the experiment and find the joint PMF of X and Y