Find the cumulative density function for the given probability density function. 1 f(x)=, for 6 ≤x≤9 F(x)=, for 6≤x≤9
Find the cumulative density function for the given probability density function. 1 f(x)=, for 6 ≤x≤9 F(x)=, for 6≤x≤9
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
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![## Cumulative Density Function
In this exercise, we will find the cumulative density function (CDF) for a given probability density function (PDF).
### Probability Density Function (PDF)
The PDF provided is:
\[ f(x) = \frac{1}{3}, \quad \text{for } 6 \leq x \leq 9 \]
### Task
We need to find the cumulative density function (CDF) \( F(x) \) for the range \( 6 \leq x \leq 9 \).
### Solution
To find the CDF \( F(x) \), we integrate the PDF \( f(x) \) from the lower limit of \( x \) to the point \( x \).
The CDF \( F(x) \) is given by the integral of \( f(t) \) from the lower bound of the interval to some value \( x \):
\[ F(x) = \int_{a}^{x} f(t) \, dt \]
For the given PDF, we integrate from 6 to \( x \):
\[ F(x) = \int_{6}^{x} \frac{1}{3} \, dt \]
Solving the integral:
\[ F(x) = \left[ \frac{1}{3}t \right]_{6}^{x} \]
\[ F(x) = \frac{1}{3}x - \frac{1}{3}(6) \]
\[ F(x) = \frac{1}{3}x - 2 \]
Thus, the cumulative density function \( F(x) \) for \( 6 \leq x \leq 9 \) is:
\[ F(x) = \frac{1}{3}x - 2, \quad \text{for } 6 \leq x \leq 9 \]
You should replace the placeholder box in the original image with:
\[ \boxed{\frac{1}{3}x - 2} \]
This method will provide the cumulative probability up to any point \( x \) within the specified interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8bde779e-c68e-4dd4-a0b8-70b65f5ae251%2F6ece6754-9e57-4c63-9d52-8b79eb03c427%2Feni79lk_processed.png&w=3840&q=75)
Transcribed Image Text:## Cumulative Density Function
In this exercise, we will find the cumulative density function (CDF) for a given probability density function (PDF).
### Probability Density Function (PDF)
The PDF provided is:
\[ f(x) = \frac{1}{3}, \quad \text{for } 6 \leq x \leq 9 \]
### Task
We need to find the cumulative density function (CDF) \( F(x) \) for the range \( 6 \leq x \leq 9 \).
### Solution
To find the CDF \( F(x) \), we integrate the PDF \( f(x) \) from the lower limit of \( x \) to the point \( x \).
The CDF \( F(x) \) is given by the integral of \( f(t) \) from the lower bound of the interval to some value \( x \):
\[ F(x) = \int_{a}^{x} f(t) \, dt \]
For the given PDF, we integrate from 6 to \( x \):
\[ F(x) = \int_{6}^{x} \frac{1}{3} \, dt \]
Solving the integral:
\[ F(x) = \left[ \frac{1}{3}t \right]_{6}^{x} \]
\[ F(x) = \frac{1}{3}x - \frac{1}{3}(6) \]
\[ F(x) = \frac{1}{3}x - 2 \]
Thus, the cumulative density function \( F(x) \) for \( 6 \leq x \leq 9 \) is:
\[ F(x) = \frac{1}{3}x - 2, \quad \text{for } 6 \leq x \leq 9 \]
You should replace the placeholder box in the original image with:
\[ \boxed{\frac{1}{3}x - 2} \]
This method will provide the cumulative probability up to any point \( x \) within the specified interval.
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