A binary erasure channel is shown in the following figure. It has two inputs {0,1} and three outputs {0,1,?}.
The capacity of the channel in terms of p is
1−p
1
p
p/2
To determine the capacity of a binary erasure channel in terms of p, we need to understand the characteristics of the channel and how it functions.
In a binary erasure channel, there are two possible inputs, 0 and 1. However, the channel can produce three possible outputs, 0, 1, or ?. The "?" represents an erasure, which means that the channel cannot determine the value of the input and it gets lost in transmission.
Now, to calculate the capacity of the channel, let's consider the probabilities of the different outcomes.
Let p be the probability of a successful transmission (output being 0 or 1) and q be the probability of an erasure (output being ?). Therefore, the probability of a successful transmission is p, and the probability of an erasure is q.
Now, since the channel is a binary erasure channel, the sum of p and q must equal 1 since they are the only possible outcomes.
p + q = 1
In terms of p, we can express q as q = 1 - p.
Now, the capacity of the channel is the maximum rate of information transfer (in bits per channel use) that can be achieved with an arbitrarily small error probability. In the case of a binary erasure channel, the maximum achievable rate is the probability of successful transmission (p) since erasures do not convey any information.
Therefore, the capacity of the binary erasure channel in terms of p is p. So, the correct answer is p.