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
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![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.
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