/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 DEMOGRAPHICS A study recently co... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 25

DEMOGRAPHICS A study recently commissioned by the mayor of a large city indicates that the number of years a current resident will continue to live in the city may be modeled as an exponential random variable with probability density function $$ f(t)= \begin{cases}0.4 e^{-0.4 t} & \text { for } t \geq 0 \\ 0 & \text { otherwise }\end{cases} $$ a. Find the probability that a randomly selected resident will move within 10 years. b. Find the probability that a randomly selected resident will remain in the city for more than 20 years. c. How long should a randomly selected resident be expected to remain in town?

Short Answer

Expert verified
a. 0.9817 b. 0.0003 c. 2.5 years

Step by step solution

01

Identify the Cumulative Distribution Function (CDF)

To solve for probabilities, express the given probability density function (PDF) as the cumulative distribution function (CDF), which is the integral of the PDF from 0 to t. The PDF is given by: \[ f(t) = 0.4 e^{-0.4 t} \text{ for } t \geq 0 \]The CDF, F(t), is: \[ F(t) = \int_{0}^{t} 0.4 e^{-0.4 x} dx \]
02

Compute the Integral

Compute the integral to find the CDF. \[ F(t) = \int_{0}^{t} 0.4 \, e^{-0.4 x} \, dx \]Perform the integration: \[ F(t) = -e^{-0.4 x} \big|_{0}^{t} \]Evaluating the definite integral gives: \[ F(t) = -e^{-0.4 t} - (-e^{0}) \]\[ F(t) = 1 - e^{-0.4 t} \]
03

Find P(T \leq 10) for part (a)

For part (a), find the probability that a randomly selected resident will move within 10 years: \[ P(T \leq 10) = F(10) = 1 - e^{-0.4 \cdot 10} \]Calculate the value: \[ P(T \leq 10) = 1 - e^{-4} \approx 1 - 0.0183 = 0.9817 \]
04

Find P(T > 20) for part (b)

For part (b), find the probability that a randomly selected resident will remain in the city for more than 20 years: \[ P(T > 20) = 1 - P(T \leq 20) \]Using the CDF, compute: \[ P(T \leq 20) = F(20) = 1 - e^{-0.4 \cdot 20} \]\[ P(T \leq 20) = 1 - e^{-8} \approx 1 - 0.0003 = 0.9997 \]\[ P(T > 20) = 1 - 0.9997 = 0.0003 \]
05

Expected value for part (c)

For part (c), calculate the expected value of the exponential distribution. The mean or expected value of an exponential random variable with rate parameter \( \lambda \) is given by: \[ E(T) = \frac{1}{\lambda} \]Here, \( \lambda = 0.4 \), so: \[ E(T) = \frac{1}{0.4} = 2.5 \text{ years} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Density Function
The Probability Density Function (PDF) represents the likelihood of a random variable taking on a specific value. In our context, this function describes the probability that a city's resident will remain for a certain number of years. The given PDF for the problem is:
\[ f(t) = 0.4 e^{-0.4 t}, \text{ for } t \text{ } \textgreater{} 0 \]
This means that the chance of a resident staying in the city decreases exponentially over time. One key feature of the PDF is that its integral over the entire possible range (from 0 to infinity) equals 1, ensuring that the total probability across all possible values sums up to 100%.
Another important point is that the PDF value can be greater than 1 in some areas, but when integrated over intervals, it gives the probability which will always be between 0 and 1.
Cumulative Distribution Function
The Cumulative Distribution Function (CDF) helps us find the probability that a resident will leave the city within a certain period. Essentially, the CDF is the area under the PDF curve from time 0 to time t. For this exercise, we need to calculate the CDF by integrating the PDF. To find the CDF, we compute:
\[ F(t) = \text{Integral}_{0}^{t} 0.4 e^{-0.4x} dx \]
Performing this integration, we get:
\[ F(t) = -e^{-0.4x} \big|_{0}^{t} = 1 - e^{-0.4t} \]
This function now gives us the probability of a resident leaving the city within time t. When we plug in different values of t, we can find specific probabilities. For example, to find the probability of leaving within 10 years, we substitute t = 10:
\[ P(T \leq 10) = F(10) = 1 - e^{-0.4 \cdot 10} \ approximate 0.9817 \]
Similarly, for staying more than 20 years:
\[ P(T > 20) = 1 - P(T \leq 20) = e^{-8} \approx 0.0003 \]
By understanding the CDF, we learn how to calculate the probability for any given time period.
Expected Value
The Expected Value (mean) of an exponential distribution tells us the average time a person is expected to remain in the city. For an exponential distribution, the formula for the expected value is given by: \[ E(T) = \frac{1}{\text{rate}} = \frac{1}{\text{\lambda}} \]
In our exercise, the rate parameter \( \lambda \) is 0.4. Hence, the expected residency time is:
\[ E(T) = \frac{1}{0.4} = 2.5 \text{ years} \]
This means that on average, a resident is expected to stay in the city for about 2.5 years. This metric is crucial for city planners and policymakers to forecast population changes and plan resources accordingly.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

PUBLIC HEALTH As part of a campaign to combat a new strain of influenza, public health authorities are planning to inoculate 1 million people. It is estimated that the probability of an individual having a bad reaction to the vaccine is \(0.0005\). Suppose the number of people inoculated who have bad reactions to the vaccine is modeled by a random variable with a Poisson distribution. a. What is \(\lambda\) for the distribution? b. What is the probability that of the 1 million people inoculated, exactly five will have a bad reaction? c. What is the probability that of the 1 million people inoculated more than 10 will have a bad reaction?

ACADEMIC TESTING The results of a calculus exam are normally distributed with a mean of \(72.3\) and a standard deviation of \(16.4\). Find the probability that a randomly chosen student's score is between 50 and 75 . If there are 82 students in the class, about how many have scores between 50 and \(75 ?\)

LOTTERY The probability of winning \(\$ 100\) in a particular lottery is \(0.08\), the probability of winning \(\$ 20\) is \(0.12\), the probability of winning \(\$ 5\) is \(0.2\), and the probability of losing is \(0.6\). What is a fair price to pay for a lottery ticket?

TRAFFIC MANAGEMENT The distance (in feet) between successive cars on a freeway is modeled by the random variable \(X\) with probability density function $$ f(x)= \begin{cases}0.25 x e^{-x / 2} & \text { if } x \geq 0 \\ 0 & \text { otherwise }\end{cases} $$ a. Find the probability that a randomly selected pair of cars will be less than 10 feet apart. b. What is the average distance between successive cars on the freeway?

If the random variable \(X\) is normally distributed with mean \(\mu=7\) and standard deviation \(\sigma=2\), what is \(P(X \geq 9)\) ?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.