Find the cumulative density function for the given probability density function. -1.7x k(x) = 1.7 e , for 0
Find the cumulative density function for the given probability density function. -1.7x k(x) = 1.7 e , for 0
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![### Finding the Cumulative Density Function (CDF)
Given the probability density function (PDF):
\[ k(x) = 1.7 e^{-1.7x}, \quad \text{for } 0 \leq x < \infty \]
We are to find the cumulative density function (CDF) \( K(x) \).
### Probability Density Function (PDF)
The PDF \( k(x) \) is given as:
\[ k(x) = 1.7 e^{-1.7x}, \quad \text{for } 0 \leq x < \infty \]
### Cumulative Density Function (CDF)
The cumulative density function \( K(x) \) is defined as the integral of the PDF from \(-\infty\) to \(x\):
\[ K(x) = \int_{-\infty}^{x} k(t) \, dt \]
Since \( k(x) \) is defined for \( 0 \leq x < \infty \), we compute the integral from 0 to \(x\):
\[ K(x) = \int_{0}^{x} 1.7 e^{-1.7t} \, dt \]
The goal is to find this integral and express it as \( K(x) \).
\[ K(x) = \left[ -e^{-1.7t} \right]_{0}^{x} \]
Evaluating this, we get:
\[ K(x) = -e^{-1.7x} + e^0 \]
\[ K(x) = -e^{-1.7x} + 1 \]
Thus, the cumulative density function is:
\[ K(x) = 1 - e^{-1.7x}, \quad \text{for } 0 \leq x < \infty \]
### Final Result
The cumulative density function \( K(x) \) for the given probability density function is:
\[ K(x) = 1 - e^{-1.7x}, \quad \text{for } 0 \leq x < \infty \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc56b3bff-26b7-4f28-9b8d-5c65e8bb1b58%2Fb9142afc-e7bc-45c1-9e89-1790e1aba87d%2Fs7l7hzg_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Cumulative Density Function (CDF)
Given the probability density function (PDF):
\[ k(x) = 1.7 e^{-1.7x}, \quad \text{for } 0 \leq x < \infty \]
We are to find the cumulative density function (CDF) \( K(x) \).
### Probability Density Function (PDF)
The PDF \( k(x) \) is given as:
\[ k(x) = 1.7 e^{-1.7x}, \quad \text{for } 0 \leq x < \infty \]
### Cumulative Density Function (CDF)
The cumulative density function \( K(x) \) is defined as the integral of the PDF from \(-\infty\) to \(x\):
\[ K(x) = \int_{-\infty}^{x} k(t) \, dt \]
Since \( k(x) \) is defined for \( 0 \leq x < \infty \), we compute the integral from 0 to \(x\):
\[ K(x) = \int_{0}^{x} 1.7 e^{-1.7t} \, dt \]
The goal is to find this integral and express it as \( K(x) \).
\[ K(x) = \left[ -e^{-1.7t} \right]_{0}^{x} \]
Evaluating this, we get:
\[ K(x) = -e^{-1.7x} + e^0 \]
\[ K(x) = -e^{-1.7x} + 1 \]
Thus, the cumulative density function is:
\[ K(x) = 1 - e^{-1.7x}, \quad \text{for } 0 \leq x < \infty \]
### Final Result
The cumulative density function \( K(x) \) for the given probability density function is:
\[ K(x) = 1 - e^{-1.7x}, \quad \text{for } 0 \leq x < \infty \]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman