Question

A radio amateur wishes to transmit a message. The frequency on which she sends the Morse signals is subject to random disturbances according to a Poisson process with intensity \(\lambda\) per second. In order to succeed with the transmission, she needs a time period of \(a\) seconds without disturbances. She stops as soon as she is done. Let \(T\) be the total time required to finish. Determine \(E T\).

Short answer

The expected total time required for a successful transmission is \( E[T] = e^{\lambda a} \).

Step-by-step solution

Understanding the Problem

We need to determine the expected time \( E[T] \) for a radio amateur to successfully transmit a message without disturbances. This relates to a Poisson process where disturbances occur at a rate \( \lambda \) per second. The transmission is only successful if there is a period of \( a \) seconds without any disturbances.

Define the Random Variables

Let \( T \) be the total time required for a successful transmission. The random disturbances follow a Poisson process with parameter \( \lambda \), meaning the time between disturbances follows an exponential distribution with mean \( \frac{1}{\lambda} \).

Compute the Probability of Success within a Given Time Frame

The probability of having no disturbances in \( a \) seconds is given by the Poisson probability formula: \( P(N(t) = 0) = e^{-\lambda t} \). Substituting \( t = a \), we get \( P(\text{no disturbances in } a \text{ seconds}) = e^{-\lambda a} \).

Derive the Expectation for Successful Transmission

The time \( T \) is a geometric random variable with success probability \( e^{-\lambda a} \). The expected value of a geometric random variable with success probability \( p \) is \( \frac{1}{p} \). Thus, the expected total time \( E[T] = \frac{1}{e^{-\lambda a}} = e^{\lambda a} \).

Conclusion

We have determined from the geometric distribution that the expected total time required for a successful Morse code transmission is \( E[T] = e^{\lambda a} \).

Key concepts

Exponential Distribution

Exponential distribution is a continuous probability distribution often used to model the time until an event occurs. In the context of the original problem, it helps to describe the time between random disturbances in a Poisson process.
The key feature of an exponential distribution is that it is memoryless. This means that the probability of an event occurring in the next time interval does not depend on how much time has already elapsed without the event occurring. This property simplifies calculations and is pivotal in analyzing processes like the one described in the problem where random disturbances affect message transmission.
The exponential distribution is characterized by a parameter \( \lambda \), which represents the rate of the process. The mean, or expected time until the next event, is \( \frac{1}{\lambda} \), making it handy for understanding the average duration between events. In our exercise, this duration corresponds to the time between disturbances the radio amateur must avoid to successfully transmit the message.

Geometric Random Variable

A geometric random variable is a discrete random variable that models the number of trials needed until the first success in a series of independent Bernoulli trials, where each trial has the same probability of success.
In the problem, the transmission success is defined as having a period of \( a \) seconds without disturbances, with each second having a probability of being disturbance-free as \( e^{-\lambda a} \). Thus, the total time \( T \) to achieve a successful transmission can be seen as a geometric random variable.
Geometric random variables have simple yet powerful properties. For instance, they capture the expected number of trials needed to get that first success, which is calculated using \( \frac{1}{p} \), where \( p \) is the probability of success. For the radio amateur's scenario, this is directly applicable as \( \frac{1}{e^{-\lambda a}} \), effectively capturing the expected time needed for an uninterrupted transmission.

Expected Value

Expected value is the average or mean value calculated for a random variable that represents the long-run average if an experiment is repeated many times.
In the exercise, we determine the expected time \( E[T] \) required for the radio amateur to successfully send her Morse code message. This is crucial because it gives a practical measure for planning purposes, namely how long transmissions can typically take given the random disturbances.
The expected value of a geometric random variable is calculated via \( \frac{1}{p} \), where \( p \) is the probability of success. Using the solution steps, we derive that \( E[T] = e^{\lambda a} \) for our Morse code transmission problem. This formula accounts for the need for \( a \) uninterrupted seconds and highlights a crucial reliance on understanding exponential distribution and successes over multiple trials, it provides helpful insights into how frequently our desired 'success' timeline can realistically be expected under specified conditions.
Expected value is not just a number—it represents the balance point of a probability distribution, offering a snapshot of expected outcomes over a long duration, which is essential for analysis and decision-making, particularly in random processes such as the one described in this problem.