Find the cumulative density function for the given probability density function. 1 f(x) = 4 for 4 ≤x≤8 F(x)=, for 4 ≤x≤ 8
Find the cumulative density function for the given probability density function. 1 f(x) = 4 for 4 ≤x≤8 F(x)=, for 4 ≤x≤ 8
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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![**Finding the Cumulative Density Function from a Given Probability Density Function**
In this example, you are given the probability density function (PDF) \( f(x) = \frac{1}{4} \) for the interval \( 4 \leq x \leq 8 \). The task is to find the corresponding cumulative density function (CDF), \( F(x) \), for the same interval.
Given:
\[ f(x) = \frac{1}{4}, \quad \text{for } 4 \leq x \leq 8 \]
To find the cumulative density function \( F(x) \), you need to integrate the probability density function \( f(x) \). The cumulative density function is defined as:
\[ F(x) = \int_{-\infty}^{x} f(t) \, dt \]
Since \( f(x) \) is zero outside the interval \( [4, 8] \), we integrate from 4 to \( x \) for \( 4 \leq x \leq 8 \):
\[ F(x) = \int_{4}^{x} \frac{1}{4} \, dt \]
Calculating the integral:
\[ F(x) = \frac{1}{4} \int_{4}^{x} \, dt \]
\[ F(x) = \frac{1}{4} [t]_{4}^{x} \]
\[ F(x) = \frac{1}{4} (x - 4) \]
Thus, the cumulative density function \( F(x) \) for \( 4 \leq x \leq 8 \) is:
\[ F(x) = \frac{1}{4} (x - 4), \quad \text{for } 4 \leq x \leq 8 \]
Entering this into the box provided in the original image gives:
\[ \boxed{\frac{1}{4} (x - 4)}, \quad \text{for } 4 \leq x \leq 8 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc56b3bff-26b7-4f28-9b8d-5c65e8bb1b58%2Fbf8c2218-3aa7-46a6-8698-0cc64d62afb9%2Fucr4hg_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding the Cumulative Density Function from a Given Probability Density Function**
In this example, you are given the probability density function (PDF) \( f(x) = \frac{1}{4} \) for the interval \( 4 \leq x \leq 8 \). The task is to find the corresponding cumulative density function (CDF), \( F(x) \), for the same interval.
Given:
\[ f(x) = \frac{1}{4}, \quad \text{for } 4 \leq x \leq 8 \]
To find the cumulative density function \( F(x) \), you need to integrate the probability density function \( f(x) \). The cumulative density function is defined as:
\[ F(x) = \int_{-\infty}^{x} f(t) \, dt \]
Since \( f(x) \) is zero outside the interval \( [4, 8] \), we integrate from 4 to \( x \) for \( 4 \leq x \leq 8 \):
\[ F(x) = \int_{4}^{x} \frac{1}{4} \, dt \]
Calculating the integral:
\[ F(x) = \frac{1}{4} \int_{4}^{x} \, dt \]
\[ F(x) = \frac{1}{4} [t]_{4}^{x} \]
\[ F(x) = \frac{1}{4} (x - 4) \]
Thus, the cumulative density function \( F(x) \) for \( 4 \leq x \leq 8 \) is:
\[ F(x) = \frac{1}{4} (x - 4), \quad \text{for } 4 \leq x \leq 8 \]
Entering this into the box provided in the original image gives:
\[ \boxed{\frac{1}{4} (x - 4)}, \quad \text{for } 4 \leq x \leq 8 \]
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