8 in images
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
#8 in images
![## Transcription for Educational Content
### Problem 6:
Let \( X_{1}, X_{2}, \ldots, X_{n} \) be the order statistics of \( n \) independent random variables \( X_{i}, X_{2}, \ldots, X_{n} \) with common probability density function (PDF) \( f(x) \). If \( 0 < x < 1 \) then
\[
f(x) = 1 \cdot e^{-x}, \quad x \geq 0
\]
Find \( E[X_{m}] \) in the following cases:
- (a) \( X_{i} \) have the common distribution function (DF)
\[
F(x) = x, \quad 0 < x < 1
\]
- (b) \( X_{i} \) have the common DF
\[
F(x) = x^{\theta}, \quad 0 < x < 1
\]
### Problem 7:
For the PDF in Problem 4, find \( E[X_{m}] \) and PDFs of \( X_{1}, X_{2}, \ldots, Y \).
\[
Y = X_{(n)} - X_{(n-1)}, Z = X_{(n)} / X_{(n-1)}
\]
### Problem 8:
An urn contains \( N \) identical marbles numbered 1 through \( N \). From the urn \( n \) marbles are drawn and let \( X_{(n)} \) be the largest number drawn. Show that
\[
P(X_{(n)} = k) = \binom{k-1}{n-1} / \binom{N}{n}, \quad k = n, n+1, \ldots, N
\]
and \( E[X_{(n)}] = \sum (N+1)/(n+1) \).
### Explanation:
This section discusses statistical problems involving probability density functions (PDFs), order statistics, and distributions.
- **Order Statistics:** In Problem 6, order statistics are used to determine the characteristics of samples drawn from independent random variables. The goal is to find the expected value (\( E[X_{m}] \)) for different cases of common distribution function.
- **Probability Functions:** The problems involve manipulating probability expressions to determine the distribution of](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3f1ebd08-7e04-46cf-88d9-0b90774940c7%2F41eb9caa-56f0-422c-86c7-18f9a995115b%2F65eifjj.jpeg&w=3840&q=75)
Transcribed Image Text:## Transcription for Educational Content
### Problem 6:
Let \( X_{1}, X_{2}, \ldots, X_{n} \) be the order statistics of \( n \) independent random variables \( X_{i}, X_{2}, \ldots, X_{n} \) with common probability density function (PDF) \( f(x) \). If \( 0 < x < 1 \) then
\[
f(x) = 1 \cdot e^{-x}, \quad x \geq 0
\]
Find \( E[X_{m}] \) in the following cases:
- (a) \( X_{i} \) have the common distribution function (DF)
\[
F(x) = x, \quad 0 < x < 1
\]
- (b) \( X_{i} \) have the common DF
\[
F(x) = x^{\theta}, \quad 0 < x < 1
\]
### Problem 7:
For the PDF in Problem 4, find \( E[X_{m}] \) and PDFs of \( X_{1}, X_{2}, \ldots, Y \).
\[
Y = X_{(n)} - X_{(n-1)}, Z = X_{(n)} / X_{(n-1)}
\]
### Problem 8:
An urn contains \( N \) identical marbles numbered 1 through \( N \). From the urn \( n \) marbles are drawn and let \( X_{(n)} \) be the largest number drawn. Show that
\[
P(X_{(n)} = k) = \binom{k-1}{n-1} / \binom{N}{n}, \quad k = n, n+1, \ldots, N
\]
and \( E[X_{(n)}] = \sum (N+1)/(n+1) \).
### Explanation:
This section discusses statistical problems involving probability density functions (PDFs), order statistics, and distributions.
- **Order Statistics:** In Problem 6, order statistics are used to determine the characteristics of samples drawn from independent random variables. The goal is to find the expected value (\( E[X_{m}] \)) for different cases of common distribution function.
- **Probability Functions:** The problems involve manipulating probability expressions to determine the distribution of
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.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