Find the cumulative density function for the given probability density function. 1 f(x) = - for 3 ≤x≤7 F(x)=, for 3≤x≤7 ...
Find the cumulative density function for the given probability density function. 1 f(x) = - for 3 ≤x≤7 F(x)=, for 3≤x≤7 ...
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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![### Finding the Cumulative Density Function
**Problem Statement:**
Find the cumulative density function for the given probability density function.
\[ f(x) = \frac{1}{4}, \quad \text{for} \quad 3 \leq x \leq 7 \]
----
The blank area represents where the cumulative density function, \( F(x) \), should be written.
**Solution:**
To find the cumulative density function \( F(x) \), we need to integrate the probability density function (pdf) \( f(x) \).
Given the probability density function:
\[ f(x) = \frac{1}{4}, \quad \text{for} \quad 3 \leq x \leq 7 \]
Integrate the pdf over the interval from the lower bound (3) to \( x \):
\[ F(x) = \int_{3}^{x} f(t) \, dt \]
Substitute \( f(t) \) with \(\frac{1}{4}\):
\[ F(x) = \int_{3}^{x} \frac{1}{4} \, dt \]
\[ F(x) = \frac{1}{4} \int_{3}^{x} 1 \, dt \]
\[ F(x) = \frac{1}{4} \left[ t \right]_{3}^{x} \]
\[ F(x) = \frac{1}{4} (x - 3) \]
Therefore, the cumulative density function \( F(x) \) is:
\[ F(x) = \frac{x - 3}{4}, \quad \text{for} \quad 3 \leq x \leq 7 \]
Below, you should input the cumulative density function in the provided solution box:
\[ F(x) = \boxed{\frac{x - 3}{4}}, \quad \text{for} \quad 3 \leq x \leq 7 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc56b3bff-26b7-4f28-9b8d-5c65e8bb1b58%2Fb13eef24-ef42-4d38-bd76-f1e9b24f59a4%2Fuuvsmui_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Cumulative Density Function
**Problem Statement:**
Find the cumulative density function for the given probability density function.
\[ f(x) = \frac{1}{4}, \quad \text{for} \quad 3 \leq x \leq 7 \]
----
The blank area represents where the cumulative density function, \( F(x) \), should be written.
**Solution:**
To find the cumulative density function \( F(x) \), we need to integrate the probability density function (pdf) \( f(x) \).
Given the probability density function:
\[ f(x) = \frac{1}{4}, \quad \text{for} \quad 3 \leq x \leq 7 \]
Integrate the pdf over the interval from the lower bound (3) to \( x \):
\[ F(x) = \int_{3}^{x} f(t) \, dt \]
Substitute \( f(t) \) with \(\frac{1}{4}\):
\[ F(x) = \int_{3}^{x} \frac{1}{4} \, dt \]
\[ F(x) = \frac{1}{4} \int_{3}^{x} 1 \, dt \]
\[ F(x) = \frac{1}{4} \left[ t \right]_{3}^{x} \]
\[ F(x) = \frac{1}{4} (x - 3) \]
Therefore, the cumulative density function \( F(x) \) is:
\[ F(x) = \frac{x - 3}{4}, \quad \text{for} \quad 3 \leq x \leq 7 \]
Below, you should input the cumulative density function in the provided solution box:
\[ F(x) = \boxed{\frac{x - 3}{4}}, \quad \text{for} \quad 3 \leq x \leq 7 \]
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