Problem 25 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

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?

Step-by-Step Solution

Answer

a. 0.9817 b. 0.0003 c. 2.5 years

Step 1: 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: \[ F(t) = \int_{0}^{t} 0.4 e^{-0.4 x} dx \]

Step 2: 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: \[ F(t) = -e^{-0.4 t} - (-e^{0}) \]\[ F(t) = 1 - e^{-0.4 t} \]

Step 3: 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: \[ P(T \leq 10) = 1 - e^{-4} \approx 1 - 0.0183 = 0.9817 \]

Step 4: 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: \[ P(T \leq 20) = F(20) = 1 - e^{-0.4 \cdot 20} \]\[ P(T \leq 20) = 1 - e^{-8} \approx 1 - 0.0003 = 0.9997 \]\[ P(T > 20) = 1 - 0.9997 = 0.0003 \]

Step 5: 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: \[ E(T) = \frac{1}{0.4} = 2.5 \text{ years} \]

Key Concepts

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:
\[ f(t) = 0.4 e^{-0.4 t}, \text{ for } t \underline{\phantom{xxx}} \textgreater{} 0 \]
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.