(a) If you have already been waiting for 10 minutes (i.e. it is 12:10pm) and the first bus has not yet come, what is the probability that you will have to wait at least 15 additional minutes? (b) Repeat part (a) under the assumption that the number of minutes from noon until the first bus is uniformly distributed over [0, 40]. (c) Explain how the answers to (a) and (b) relate to whether exponential and uniform random variables are memoryless.
(a) If you have already been waiting for 10 minutes (i.e. it is 12:10pm) and the first bus has not yet come, what is the probability that you will have to wait at least 15 additional minutes? (b) Repeat part (a) under the assumption that the number of minutes from noon until the first bus is uniformly distributed over [0, 40]. (c) Explain how the answers to (a) and (b) relate to whether exponential and uniform random variables are memoryless.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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![**Title: Understanding Exponential and Uniform Random Variables in Real-Life Scenarios**
**Scenario:**
One day at noon, you have a sudden craving for a California burrito from the Taco Stand, and you decide to take the bus from campus into town. Starting at noon, suppose that the number of minutes until the first bus arrives is an exponential random variable with parameter \( \frac{1}{20} \).
**Questions:**
(a) If you have already been waiting for 10 minutes (i.e., it is 12:10 pm) and the first bus has not yet come, what is the probability that you will have to wait at least 15 additional minutes?
(b) Repeat part (a) under the assumption that the number of minutes from noon until the first bus is uniformly distributed over \([0, 40]\).
(c) Explain how the answers to (a) and (b) relate to whether exponential and uniform random variables are memoryless.
**Analysis and Explanation:**
- **Exponential Random Variable:**
- Exponential variables are defined by a constant rate parameter, in this case, \( \frac{1}{20} \). This characterizes the average waiting time, making the exponential variable memoryless. The probability of waiting for an additional time period is the same, regardless of how much time has already passed without the event occurring.
- **Uniform Random Variable:**
- A uniform random variable is evenly distributed across a defined range, \([0, 40]\) in this case, meaning every minute within this range has an equal probability of the bus arriving. Unlike the exponential variable, it does not possess the memoryless property.
This problem examines the concept of memorylessness and compares how exponential and uniform distributions behave in a practical setting.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25ecc335-25be-4ebd-8f8c-629ee64b2629%2F9b59deed-e328-4d01-bf93-79650dcdb327%2Fbdyyvk_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Exponential and Uniform Random Variables in Real-Life Scenarios**
**Scenario:**
One day at noon, you have a sudden craving for a California burrito from the Taco Stand, and you decide to take the bus from campus into town. Starting at noon, suppose that the number of minutes until the first bus arrives is an exponential random variable with parameter \( \frac{1}{20} \).
**Questions:**
(a) If you have already been waiting for 10 minutes (i.e., it is 12:10 pm) and the first bus has not yet come, what is the probability that you will have to wait at least 15 additional minutes?
(b) Repeat part (a) under the assumption that the number of minutes from noon until the first bus is uniformly distributed over \([0, 40]\).
(c) Explain how the answers to (a) and (b) relate to whether exponential and uniform random variables are memoryless.
**Analysis and Explanation:**
- **Exponential Random Variable:**
- Exponential variables are defined by a constant rate parameter, in this case, \( \frac{1}{20} \). This characterizes the average waiting time, making the exponential variable memoryless. The probability of waiting for an additional time period is the same, regardless of how much time has already passed without the event occurring.
- **Uniform Random Variable:**
- A uniform random variable is evenly distributed across a defined range, \([0, 40]\) in this case, meaning every minute within this range has an equal probability of the bus arriving. Unlike the exponential variable, it does not possess the memoryless property.
This problem examines the concept of memorylessness and compares how exponential and uniform distributions behave in a practical setting.
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