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.
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?
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
1.p*q
2.No
1. The new process Zn is not Bernoulli with a fixed parameter. The parameter of Zn depends on the parameters of Xn and Yn, denoted by p and q, respectively. The probability of a success in Zn is given by the product of the probabilities of successes in Xn and Yn, i.e., p*q.
2. If the two Bernoulli processes Xn and Yn are dependent, the process Zn is not guaranteed to be Bernoulli. The independence assumption between the pairs (Xn, Yn) does not ensure that the product of Xn and Yn will yield a Bernoulli process. The dependence between Xn and Yn can introduce additional complexities that make Zn's distribution more complicated than a simple Bernoulli distribution. So, it is not guaranteed to be Bernoulli. But hey, at least there's some uncertainty in life, right? Adds a little excitement!
1. To determine the parameter of the new process Zn, we need to analyze the relationship between the parameters of the original processes Xn and Yn.
Since Xn and Yn are independent Bernoulli processes, each with its own parameter, we can express the probability of an arrival in a given time slot as follows:
P(Xn = 1) = p
P(Yn = 1) = q
To form the new process Zn, an arrival is recorded if and only if both Xn and Yn record an arrival in that same time slot. This means that for Zn to be equal to 1, both Xn and Yn must be equal to 1:
P(Zn = 1) = P(Xn = 1 and Yn = 1)
Since Xn and Yn are independent, the probability of the intersection of events is equal to the product of their individual probabilities:
P(Zn = 1) = P(Xn = 1) * P(Yn = 1) = p * q
Therefore, the new process Zn is also Bernoulli with a parameter equal to the product of the parameters of Xn and Yn, i.e., Zn ~ Bernoulli(p * q).
2. In this case, where Xn and Yn are dependent but the pairs (Xn,Yn) are independent, the process Zn is not guaranteed to be Bernoulli.
To illustrate this, let's consider an example. Assume Xn and Yn follow the following patterns:
Xn: 0 1 0 1 0 1 ...
Yn: 0 1 1 0 0 1 ...
Following the given pattern, we can calculate the process Zn, which is the logical AND operation between Xn and Yn:
Zn: 0 1 0 0 0 1 ...
As we can see from this example, Zn does not follow a Bernoulli distribution since it does not have a constant parameter. In Bernoulli processes, the parameter remains constant throughout the process, but in this case, the parameter is not fixed as it depends on the specific values of Xn and Yn in each time slot.
Therefore, if the two Bernoulli processes Xn and Yn are dependent, the process Zn is not guaranteed to be Bernoulli.
1.0
2.No