Problem 121 - Applied Statistics and Probability for Engineers

Question

The time between arrivals of taxis at a busy intersection is exponentially distributed with a mean of 10 minutes. (a) What is the probability that you wait longer than one hour for a taxi? (b) Suppose that you have already been waiting for one hour for a taxi. What is the probability that one arrives within the next 10 minutes? (c) Determine \(x\) such that the probability that you wait more than \(x\) minutes is \(0.10 .\) (d) Determine \(x\) such that the probability that you wait less than \(x\) minutes is 0.90 . (e) Determine \(x\) such that the probability that you wait less than \(x\) minutes is \(0.50 .\)

Step-by-Step Solution

Answer

(a) 0.0025; (b) 0.6321; (c) 23.03 min; (d) 23.03 min; (e) 6.93 min.

Step 1: Understanding the Exponential Distribution

The waiting time for a taxi is exponentially distributed with a mean of 10 minutes. The exponential probability density function is given by \( f(t) = \lambda e^{-\lambda t} \) where \( \lambda \) is the rate parameter. The mean of an exponential distribution is \( \frac{1}{\lambda} = 10 \) minutes, so \( \lambda = 0.1 \) per minute.

Step 2: Probability of Waiting Longer Than One Hour (Part a)

To find the probability of waiting longer than one hour (60 minutes), we use the cumulative distribution function (CDF): \( P(T > t) = e^{-\lambda t} \). Plug in \( t = 60 \) and \( \lambda = 0.1 \):\[ P(T > 60) = e^{-0.1 \times 60} = e^{-6} \approx 0.0025. \]

Step 3: Probability of Arrival Within 10 Minutes After Waiting 1 Hour (Part b)

Exponential distribution is memoryless, meaning the probability of waiting additional time is independent of past waiting time. Thus, for the next 10 minutes, use:\[ P(T \leq 10) = 1 - e^{-\lambda \times 10} = 1 - e^{-1} \approx 0.6321. \]

Step 4: Find x Such That Waiting More than x Minutes is 0.10 (Part c)

Use the CDF to find \( x \):\[ P(T > x) = 0.10 \Rightarrow e^{-\lambda x} = 0.10 \Rightarrow -\lambda x = \ln(0.10) \Rightarrow x = -\frac{\ln(0.10)}{0.1}. \]Calculate:\[ x = -\frac{\ln(0.10)}{0.1} \approx 23.03. \]

Step 5: Find x Such That Waiting Less than x Minutes is 0.90 (Part d)

Use the CDF for \( P(T \leq x) = 0.90 \):\[ 1 - e^{-\lambda x} = 0.90 \Rightarrow e^{-\lambda x} = 0.10 \Rightarrow -\lambda x = \ln(0.10) \Rightarrow x = -\frac{\ln(0.10)}{0.1}. \]Since \( P(T > x) = 0.10 \) and result from Step 4: \( x \approx 23.03. \)

Step 6: Find x Such That Waiting Less than x Minutes is 0.50 (Part e)

Use the CDF: \( P(T \leq x) = 0.50 \):\[ 1 - e^{-\lambda x} = 0.50 \Rightarrow e^{-\lambda x} = 0.50 \Rightarrow -\lambda x = \ln(0.50) \Rightarrow x = -\frac{\ln(0.50)}{0.1}. \]Calculate:\[ x = -\frac{\ln(0.50)}{0.1} \approx 6.93. \]

Key Concepts

Probability Density Function

The Probability Density Function (PDF) is a central concept in statistics, particularly in the study of continuous random variables. When dealing with an exponential distribution, the PDF helps us understand how probabilities are distributed over various possible outcomes. In the context of the exponential distribution, the PDF is expressed as:\[ f(t) = \lambda e^{-\lambda t} \]Here, the Greek letter \(\lambda\) represents the rate parameter, and \(t\) is the time variable.

• The PDF tells us the probability density of the random variable reaching a particular value. However, note that the actual probability at any single point is zero; the interest lies in an interval.
• For example, if a taxi arrives following an exponential distribution with a certain rate parameter, the PDF will shape how we expect these taxi arrivals over time.

The PDF's shape is characterized by a rapid decrease, emphasizing the probability of shorter waiting times, which is typical for events that follow an exponential distribution. This implies that there is a higher chance of observing smaller values of time before an event occurs, such as the arrival of a taxi.

Memoryless Property

One of the fascinating properties of the exponential distribution is its memoryless property. This characteristic is mathematically significant and heavily influences practical interpretations, especially in real-world applications. Such a property means that the future probability of an event that happens after a certain amount of time does not depend on what has happened before.

• In layman's terms, the probability of a taxi arriving in the next 10 minutes is the same whether you have waited for 1 minute or 1 hour.
• This is due to the mathematical fact that any section of time is independent and has the same exponential characteristic.

Mathematically, the memoryless property is denoted as it applies for any positive numbers \(s\) and \(t\):\[ P(T > s + t \mid T > s) = P(T > t) \]This unique property makes exponential distributions particularly useful for modeling wait times for events that happen continuously and independently over time, like the arrival of taxis or phone calls.

Cumulative Distribution Function

The Cumulative Distribution Function (CDF) is crucial when dealing with continuous distributions because it gives us a method to calculate probabilities not just at a point, but over an interval. The CDF of an exponential distribution provides the probability that the random variable takes a value less than or equal to a particular number. It is a central tool to solve various problems, including those involving waiting times.The CDF for the exponential distribution is given by:\[ P(T \leq t) = 1 - e^{-\lambda t} \]

• Using the CDF, we can determine the probability of waiting no more than a certain amount of time before an event occurs.
• For instance, it's useful for estimating how long you'll wait for a taxi or determining the cut-off time to catch 90% of arrivals.

The CDF ramps up quickly as \(t\) increases, reflecting the decreasing likelihood of longer wait times in exponentially distributed processes. Solving problems using the CDF involves setting it to a particular probability threshold and solving for \(t\). This is what gives rise to solutions in homework problems where you're asked to wait less than a certain amount of minutes or more.

Rate Parameter

The rate parameter \(\lambda\) is a defining characteristic of the exponential distribution. It represents how quickly or slowly events occur. Understanding \(\lambda\) is fundamental when working with exponential distributions as it directly influences both the Probability Density Function and the Cumulative Distribution Function.

• The rate parameter \(\lambda\) is inversely related to the mean of the distribution. For instance, if the mean waiting time for a taxi is 10 minutes, the rate parameter is \(\lambda = 0.1\) per minute.
• Higher \(\lambda\) values indicate more frequent occurrences of events, leading to shorter mean waiting times.

In practical terms, when the rate parameter is higher, it means you'll likely experience shorter waiting times. By influencing the shape of the distribution, \(\lambda\) aids in calculating different probabilities and making predictions about waiting times. Therefore, it plays a pivotal role in problems that require determining how long we'll wait for events, like taxis at an intersection.