Problem 120 - Applied Statistics and Probability for Engineers

Question

The time between the arrival of electronic messages at your computer is exponentially distributed with a mean of two hours. (a) What is the probability that you do not receive a message during a two- hour period? (b) If you have not had a message in the last four hours, what is the probability that you do not receive a message in the next two hours? (c) What is the expected time between your fifth and sixth messages?

Step-by-Step Solution

Answer

(a) \( e^{-1} \), (b) \( e^{-1} \), (c) 2 hours.

Step 1: Identify the Information Given

We have an exponentially distributed variable for the time between messages with a mean of 2 hours. The exponential distribution parameter \( \lambda \) is the reciprocal of the mean. Hence, \( \lambda = \frac{1}{2} \) per hour.

Step 2: Calculate the Probability for Part (a)

In part (a), we need the probability of receiving no messages within 2 hours. Using the exponential distribution, this is calculated as: \( P(T > 2) = e^{-\lambda \cdot t} \) where \( t = 2 \) hours and \( \lambda = \frac{1}{2} \). Thus, \( P(T > 2) = e^{-\frac{1}{2} \cdot 2} = e^{-1} \).

Step 3: Calculate the Probability for Part (b) Using Memoryless Property

For part (b), because of the memoryless property of exponential distributions, the probability that no message arrives in the next 2 hours given that no message has arrived in the past 4 hours is the same as if no time has passed. Thus, \( P(T > 2) \) is the same as in part (a), so \( P(T > 2) = e^{-1} \).

Step 4: Determine the Expectation for Part (c)

For part (c), exponential distributions have the memoryless property, and each message arrival is independent. The expected time between each message is constant. Therefore, the expected time between the fifth and sixth messages is equal to the mean time, which is 2 hours.

Key Concepts

Probability

Probability is a measure of how likely an event is to occur. It lies between 0 and 1, where 0 indicates an impossibility and 1 indicates certainty. In the context of the exponential distribution, we are often interested in the probability of events concerning time until a certain occurrence, such as the arrival of messages. The exponential distribution is a continuous probability distribution that is used to model time until a specific event occurs. It is particularly useful for reliability and waiting time applications.

For the exercise given, we can calculate the probability of receiving no messages in a two-hour window using the formula: \[ P(T > t) = e^{-\lambda \cdot t} \] where \( \lambda \) is the rate parameter, and \( t \) is the time interval of interest. This gives us an exponential random variable, making it possible to model time predictions. In our specific problem, this was used to find the probability that no messages are received in two hours, calculated as \( e^{-1} \).

Understanding probability within this context enables us to make informed predictions about systems governed by random arrivals or occurrences.

Memoryless Property

The memoryless property is a distinctive feature of the exponential distribution. It implies that the probability of an event occurring in the future is independent of any past events. This unique property simplifies many calculations because it means the probability of an event occurring in a future interval is the same, regardless of elapsed time.

Let's say you're waiting for a message that hasn’t arrived for four hours; the memoryless property allows us to treat the next two-hour period as if it were starting from scratch. Thus, our problem's part (b) calculation mirrors part (a) because the likelihood of receiving a message in the next two hours does not depend on the previous four hours' lack of messages. Both scenarios result in the probability \( P(T > 2) = e^{-1} \). This can be counterintuitive at first but is a powerful tool in probability theory for dealing with exponential distributions.

• Independent of past events
• Simplifies calculations
• Useful in reliability and service time analysis

Expected Value

The expected value in probability and statistics refers to the average or mean value predicted for a random variable in a distribution. It gives us an estimate of the central tendency of the distribution.

In an exponential distribution, the expected value – denoted often as \( E[T] \) – is the inverse of the rate parameter \( \lambda \). This is because the mean waiting time is \( \frac{1}{\lambda} \), where \( \lambda \) represents how often an event occurs per time unit.

In the exercise, we're tasked to find the expected time between the fifth and sixth messages, which in an exponential distribution is simply the mean time interval between events. Here, the mean is given as two hours, meaning every message interval, regardless of its sequence in line (5th to 6th or otherwise), has an expected duration of 2 hours. This consistency is beneficial in anticipatory planning and resource allocation.

• Represents the average over multiple instances
• For exponential distribution, it's the reciprocal of the rate
• Fundamental for decision-making and risk assessment

Exponential Distribution Parameter

The exponential distribution parameter, commonly denoted as \( \lambda \), plays a crucial role. It is the rate at which events occur, or, in easier terms, it is the average number of events expected in a time unit. It is the fundamental part of the exponential distribution and inversely related to the mean time between events, \( \frac{1}{\lambda} \).

In problems with an exponential distribution, knowing \( \lambda \) allows you to calculate probabilities for events over different time spans easily. In our exercise, the distribution's mean was 2 hours, translating to a \( \lambda \) of \( \frac{1}{2} \) per hour, signifying that on average, half an event (or 1 event per 2 hours) is expected. This parameter defines the shape and scale of the distribution, determining how often events happen without past influence, adhering to the memoryless property.

Comprehending \( \lambda \) is key as it provides the link to the probability and expected value calculations, relevant in fields like queuing theory and survival analysis to predict future occurrences and allocate resources effectively.

• Rate of occurrence per time unit
• Inversely proportional to the mean
• Essential for modeling randomness in timings