Question

Consider a system that can be in one of two states: "on " or "off." At time zero it is "on." It then serves before breakdown for a random time \(T_{\text {on }}\) with distribution function \(1-e^{-t \lambda}\). It is then off before being repaired for a random time \(T_{\text {oft }}\) with the same distribution funetion \(1-e^{-t \lambda}\). It then repeats a statistically independent and identically distributed similar cyele, and so on. Determine the mean of \(W(t)\), the random variable measuring the total time the system is operating during the interval \((0, t)\),

Short answer

In summary, the mean of the random variable \(W(t)\), which measures the total time the system is operating during the interval \((0, t)\), is \(\frac{1}{2}\). This means the system is expected to be "on" for half the total time during the interval \((0, t)\).

Step-by-step solution

Analyze the given distribution function

Since the "on" state and the "off" state have the same distribution function, we can denote it as \(F(t) = 1 - e^{-t \lambda}\). This is a cumulative distribution function, meaning it gives the probability that the random variable (in this case, \(T_{\text{on}}\) and \(T_{\text{off}}\)) is less than or equal to \(t\).

Find the probability density function

To find the probability density function (PDF) for both "on" and "off" states, we need to take the derivative of the cumulative distribution function with respect to \(t\). So, the PDF for \(T_{\text{on}}\) and \(T_{\text{off}}\) is given by: \(f(t) = \frac{dF(t)}{dt} = \lambda e^{-t \lambda}\)

Calculate the expected value of "on" and "off" times

To find the mean values of \(T_{\text{on}}\) and \(T_{\text{off}}\), we need to find the expected value for both random variables. Since both have the same PDF, their expected values will be the same. The expected value, \(E[T]\), is given by: \(E[T] = \int_0^{\infty} t \cdot f(t) dt = \int_0^{\infty} t \cdot \lambda e^{-t \lambda} dt\) To evaluate this integral, we can use integration by parts, where: \(u = t \implies du = dt\) \(dv = \lambda e^{-t \lambda} dt \implies v = -e^{-t \lambda}\) Now, substitute to get: \(E[T] = -te^{-t \lambda} \Big|_{0}^{\infty} + \int_0^{\infty} e^{-t \lambda} dt\) The left term in the sum is zero because both limits make either \(t=0\) or the exponential term equal zero. So, we are left with the integral: \(E[T] = \int_0^{\infty} e^{-t \lambda} dt = \frac{1}{\lambda} \Big|_{0}^{\infty} = \frac{1}{\lambda}\) Thus, the expected value of both "on" and "off" times is \(\frac{1}{\lambda}\).

Calculate the mean of \(W(t)\)

Since the duration of each "on" and "off" state is the same, the fraction of time the system is in the "on" state is \(\frac{1}{2}\). Finding the mean of \(W(t)\) is equivalent to finding the total time the system is expected to be "on" during the interval \((0, t)\). Mean of \(W(t) = \frac{\text{Expected time in "on" state}}{\text{Total time}} = \frac{\frac{1}{\lambda}}{\frac{1}{\lambda} + \frac{1}{\lambda}} = \frac{1}{2}\). So, the mean of \(W(t)\) is \(\frac{1}{2}\), which means that the system is expected to be "on" for half the total time during the interval \((0, t)\).

Key concepts

Cumulative Distribution Function

The Cumulative Distribution Function (CDF), a fundamental concept in the study of stochastic processes, is a mathematical tool that quantifies the probability that a random variable will take on a value less than or equal to a specific threshold. In our exercise, the CDF for the random times the system is in the 'on' or 'off' state is expressed by the function \(F(t) = 1 - e^{-t \lambda}\). This function gives us a complete description of the probability behavior of the system over time.

Understanding the CDF is critical because it provides a way to calculate probabilities and various statistical measures. It is always non-decreasing and bounded between 0 and 1, mapping the possible values of a random variable to their cumulative probabilities. A CDF is useful because it gives us insights into the likelihood of certain outcomes and can help predict the behavior of complex systems.

Probability Density Function

While the CDF measures cumulative probabilities, the Probability Density Function (PDF) specifies the probability of the random variable falling within a particular range of values. Essentially, it describes the density of probability at each point. For the exercise in question, the PDF, derived from the CDF, is represented as \(f(t) = \lambda e^{-t \lambda}\).

The PDF is fundamentally important because it gives us a granular view of how probabilities are distributed across different outcomes. In the case of continuous random variables, the PDF is the derivative of the CDF. It is important to note that the PDF itself is not a probability, but an infinitesimally small range around a specific value of the random variable. So to find the probability of the random variable falling within a certain interval, we would integrate the PDF over that interval.

Expected Value

The Expected Value, or mean, of a probability distribution is a measure of the central tendency, representing the average outcome we would expect if we were to repeat an experiment a large number of times. In the context of our exercise, it provides a way to predict the average time the system spends in the 'on' state by evaluating \(E[T] = \int_0^{\infty} t \cdot f(t) dt\).

Calculating the expected value involves integrating the product of the value with its probability over all possible outcomes. This not only helps to predict the future performance of the system but also establishes a foundation for more complex statistical measures, such as variance and standard deviation, which provide information about the spread or variability around the expected value.

Random Variables

Random Variables are a cornerstone of probability theory, as they enable us to quantify and analyze randomness and uncertainty in stochastic processes. They are variables whose possible values are numerical outcomes of a random phenomenon. In our system example, \(T_{\text{on}}\) and \(T_{\text{off}}\) represent the random durations for which the system is 'on' and 'off', respectively.

There are two types of random variables: discrete and continuous. Discrete random variables have a countable number of possible values, while continuous random variables can take on any value within a continuous range. The behavior of random variables is characterized by their probability distributions, such as the one given by the CDF and PDF in our exercise, allowing us to perform probabilistic calculations and gain insights into the system's behavior over time.