**Educational Website Transcription**---**Topic: Reducing Waiting Times in Supermarkets**A supermarket chain has integrated computer simulation and information technology to reduce the average waiting time for customers at 2,300 stores. By implementing a new predictive system, the average customer waiting time has improved to just 22 seconds.**Problem: Understanding Waiting Time Distribution**(a) **Exponential Distribution of Waiting Time**Assume that supermarket waiting times are exponentially distributed. Demonstrate the probability density function (pdf) of waiting time at the supermarket:\[ f(x) = \begin{cases} & \text{__________}, \quad x \ge 0 \\ & \text{__________}, \quad \text{elsewhere} \end{cases}\](b) **Probability Calculation between 30 and 45 Seconds**Calculate the probability that a customer will need to wait between 30 and 45 seconds. (Round your answer to four decimal places.)- Answer: 0.5056 ❌(c) **Probability of Waiting More than 2 Minutes**Determine the probability that a customer will have to wait more than 2 minutes. (Round your answer to four decimal places.)- Answer: 0.9140 ❌---Note: The answers provided in parts (b) and (c) are marked incorrect. Review your calculations and ensure they're based on the correct parameters and assumptions for the exponential distribution.

Given: Average waiting time of the customers λ=22 seconds

Do you dislike waiting in line? A supermarket chain has used computer simulation and information technology to reduce the average waiting time for customers at 2,300 stores. Using a new system, which allows the supermarket to better predict when shoppers will be checking out, the company was able to decrease average customer waiting time to just 22 seconds. (a) Assume that supermarket waiting times are exponentially distributed. Show the probability density function of waiting time at the supermarket. x2 0 f(x) = elsewhere (b) What is the probability that customer will have to wait between 30 and 45 seconds? (Round your answer to four decimal places.) 0.5056 (c) What is the probability that customer will have to wait more than minutes? (Round your answer to four decimal places.) 0.9140

Do you dislike waiting in line? A supermarket chain has used computer simulation and information technology to reduce the average waiting time for customers at 2,300 stores. Using a new system, which allows the supermarket to better predict when shoppers will be checking out, the company was able to decrease average customer waiting time to just 22 seconds. (a) Assume that supermarket waiting times are exponentially distributed. Show the probability density function of waiting time at the supermarket. x2 0 f(x) = elsewhere (b) What is the probability that customer will have to wait between 30 and 45 seconds? (Round your answer to four decimal places.) 0.5056 (c) What is the probability that customer will have to wait more than minutes? (Round your answer to four decimal places.) 0.9140

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Concept explainers

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!

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**Educational Website Transcription**

---

**Topic: Reducing Waiting Times in Supermarkets**

A supermarket chain has integrated computer simulation and information technology to reduce the average waiting time for customers at 2,300 stores. By implementing a new predictive system, the average customer waiting time has improved to just 22 seconds.

**Problem: Understanding Waiting Time Distribution**

(a) **Exponential Distribution of Waiting Time**

Assume that supermarket waiting times are exponentially distributed. Demonstrate the probability density function (pdf) of waiting time at the supermarket:

\[ f(x) =
\begin{cases}
& \text{__________}, \quad x \ge 0 \\
& \text{__________}, \quad \text{elsewhere}
\end{cases}
\]

(b) **Probability Calculation between 30 and 45 Seconds**

Calculate the probability that a customer will need to wait between 30 and 45 seconds. (Round your answer to four decimal places.)

- Answer: 0.5056 ❌

(c) **Probability of Waiting More than 2 Minutes**

Determine the probability that a customer will have to wait more than 2 minutes. (Round your answer to four decimal places.)

- Answer: 0.9140 ❌

---

Note: The answers provided in parts (b) and (c) are marked incorrect. Review your calculations and ensure they're based on the correct parameters and assumptions for the exponential distribution.

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