We start with two independent Bernoulli processes, Xn and Yn, with parameters p and q, respectively. We form a new process Zn by recording an arrival in a given time slot if and only if both of the original processes record an arrival in that same time slot. Mathematically, Zn = XnYn.

Question

1. The new process Zn is also Bernoulli with parameter.

2. Suppose that the two Bernoulli processes Xn and Yn are dependent. We still assume, however, that the pairs (Xn,Yn) are independent. E.g., (X1,Y1) is independent from (X2,Y2), etc. Is the process Zn guaranteed to be Bernoulli?

Answer 1

1.0
2.No

Answer 2

The merged process records an arrival if and only if both of the original processes record an arrival, which happens with probability
1.pq
2.No

Answer 3

1.p*q
2.No