Find the proportion X of individuals who can be expected to respond to a certain mail-order solicitation if X has the density function below. f(x) = 2(x + 10) 21 0, 0
Find the proportion X of individuals who can be expected to respond to a certain mail-order solicitation if X has the density function below. f(x) = 2(x + 10) 21 0, 0
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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16. Please answer the following question.
![**Probability Density Function for Mail-Order Solicitation Response**
To determine the proportion \( X \) of individuals expected to respond to a mail-order solicitation, we analyze the given probability density function (PDF):
\[
f(x) =
\begin{cases}
\frac{2(x+10)}{21}, & 0 < x < 1 \\
0, & \text{elsewhere}
\end{cases}
\]
**Calculating the Proportion**
The problem requires finding the total probability that an individual will respond, which is the integral of the PDF over its defined range:
1. **Range of Integration**: The function is defined between 0 and 1.
2. **Integrate the PDF over the interval \(0 < x < 1\)**:
\[ \int_{0}^{1} \frac{2(x+10)}{21} \, dx \]
Simplifying this integral will yield the total probability.
**Solution**
The result of the integration will be the proportion of individuals expected to respond, which should be entered as an integer or a simplified fraction.
Consider this function to reflect the behavioral response of individuals based on mail-order solicitations, highlighting the importance of using probability density functions in market analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabd5e741-0235-409f-80de-883f5f0e5da7%2Fbe8e2e73-6c4c-44e3-af63-bad5a2a26fa0%2Fo52qiw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Probability Density Function for Mail-Order Solicitation Response**
To determine the proportion \( X \) of individuals expected to respond to a mail-order solicitation, we analyze the given probability density function (PDF):
\[
f(x) =
\begin{cases}
\frac{2(x+10)}{21}, & 0 < x < 1 \\
0, & \text{elsewhere}
\end{cases}
\]
**Calculating the Proportion**
The problem requires finding the total probability that an individual will respond, which is the integral of the PDF over its defined range:
1. **Range of Integration**: The function is defined between 0 and 1.
2. **Integrate the PDF over the interval \(0 < x < 1\)**:
\[ \int_{0}^{1} \frac{2(x+10)}{21} \, dx \]
Simplifying this integral will yield the total probability.
**Solution**
The result of the integration will be the proportion of individuals expected to respond, which should be entered as an integer or a simplified fraction.
Consider this function to reflect the behavioral response of individuals based on mail-order solicitations, highlighting the importance of using probability density functions in market analysis.
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