Suppose that the amount of time a hospital patient must wait for a nurse's help is described by a continuous random variable with density function f(t) = e-t/3 where t≥ 0 is measured in minutes. (a) What is the probability that a patient must wait for more than 4 minutes? (b) A patient spends a week in the hospital and requests nurse assistance once each day. What is the probability that the nurse will take longer than 5 minutes to respond on (exactly) two occasions? (c) What is the probability that on at least one call out of seven, the nurse will take longer than 7 minutes to respond?
Suppose that the amount of time a hospital patient must wait for a nurse's help is described by a continuous random variable with density function f(t) = e-t/3 where t≥ 0 is measured in minutes. (a) What is the probability that a patient must wait for more than 4 minutes? (b) A patient spends a week in the hospital and requests nurse assistance once each day. What is the probability that the nurse will take longer than 5 minutes to respond on (exactly) two occasions? (c) What is the probability that on at least one call out of seven, the nurse will take longer than 7 minutes to respond?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![The given problem illustrates a situation where the time a hospital patient must wait for a nurse's assistance is modeled by a continuous random variable with a specific density function. The function is given by:
\[ f(t) = \frac{1}{3} e^{-t/3} \]
for \( t \geq 0 \) (time is measured in minutes).
### Questions:
**(a)** What is the probability that a patient must wait for more than 4 minutes?
To find this, you would calculate the probability \( P(T > 4) \).
**(b)** A patient spends a week in the hospital and requests nurse assistance once each day. What is the probability that the nurse will take longer than 5 minutes to respond on (exactly) two occasions?
Here, we look at a binomial distribution scenario with 7 trials (one for each day), where the probability of taking longer than 5 minutes is calculated using the given density function. We aim to find the probability of exactly two such incidents.
**(c)** What is the probability that on at least one call out of seven, the nurse will take longer than 7 minutes to respond?
This involves calculating the probability of the complement (nurse taking less than or equal to 7 minutes) for all seven calls and subtracting from 1 to find the probability of at least one over 7 minutes.
### Explanation of the Density Function:
The function \( f(t) = \frac{1}{3} e^{-t/3} \) is an exponential density function. The exponential distribution is often used to model waiting times or the time until an event occurs. Here, the rate parameter \( \lambda = \frac{1}{3} \) indicates the average rate of the event (i.e., nurse response) occurrence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1941a39f-dbc9-477a-8528-f1bffe3dd4bc%2Feb1ddb8b-613b-405b-ab0d-2f0086ce3fc3%2Fcrapzu_processed.png&w=3840&q=75)
Transcribed Image Text:The given problem illustrates a situation where the time a hospital patient must wait for a nurse's assistance is modeled by a continuous random variable with a specific density function. The function is given by:
\[ f(t) = \frac{1}{3} e^{-t/3} \]
for \( t \geq 0 \) (time is measured in minutes).
### Questions:
**(a)** What is the probability that a patient must wait for more than 4 minutes?
To find this, you would calculate the probability \( P(T > 4) \).
**(b)** A patient spends a week in the hospital and requests nurse assistance once each day. What is the probability that the nurse will take longer than 5 minutes to respond on (exactly) two occasions?
Here, we look at a binomial distribution scenario with 7 trials (one for each day), where the probability of taking longer than 5 minutes is calculated using the given density function. We aim to find the probability of exactly two such incidents.
**(c)** What is the probability that on at least one call out of seven, the nurse will take longer than 7 minutes to respond?
This involves calculating the probability of the complement (nurse taking less than or equal to 7 minutes) for all seven calls and subtracting from 1 to find the probability of at least one over 7 minutes.
### Explanation of the Density Function:
The function \( f(t) = \frac{1}{3} e^{-t/3} \) is an exponential density function. The exponential distribution is often used to model waiting times or the time until an event occurs. Here, the rate parameter \( \lambda = \frac{1}{3} \) indicates the average rate of the event (i.e., nurse response) occurrence.
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps

Recommended textbooks for you

MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc

Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning

Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning

MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc

Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning

Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning

Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON

The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman

Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman