Question

Each time a modem transmits one bit, the receiving modem analyzes the signal that arrives and decides whether the transmitted bit is o or 1. It makes an error with probability p, independent of whether any other bit is received correctly. If the transmission continues until the receiving modem makes its first error, what is the PMF of X, the number of bits transmitted?

Answer 1

The PMF of the number of bits transmitted until the first error is given by a geometric distribution P(X = x) = (1 - p)^(x-1) * p, where p is the probability of an error on a single bit.

Step-by-step explanation:

The question asks for the probability mass function (PMF) of X, representing the number of bits transmitted by a modem until the first error occurs. Since the errors occur independently with a probability p and the process continues until the first error, this scenario follows a geometric distribution. The PMF for a geometric distribution is given by P(X = x) = (1 - p)^(x-1) * p, where X is the number of trials up to and including the first success (in this case, an error), and p is the probability of success (error) on each trial.

To further elaborate, if we want to find the probability that the first error occurs on the 7th bit, we would calculate P(X = 7) using the formula, resulting in (1 - p)^6 * p.