weighs them. The results (in kg) are as follows. Weight, W Frequency Weight, W Frequency 0
weighs them. The results (in kg) are as follows. Weight, W Frequency Weight, W Frequency 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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![### Statistical Study of Trout Weights for Fish Farming Decisions
#### Problem Statement:
A fish farmer is investigating the weight distribution of trout in a lake to decide whether to net the lake and sell the fish. A sample of 100 fish was collected and weighed, yielding the following results:
#### Data:
The weights of the fish are organized in two tables, representing different weight ranges and their corresponding frequencies.
| Weight, \( W \) (kg) | Frequency |
|----------------------|-----------|
| 0 < \( W \) ≤ 0.5 | 2 |
| 0.5 < \( W \) ≤ 1.0 | 10 |
| 1.0 < \( W \) ≤ 1.5 | 23 |
| 1.5 < \( W \) ≤ 2.0 | 26 |
| Weight, \( W \) (kg) | Frequency |
|----------------------|-----------|
| 2.0 < \( W \) ≤ 2.5 | 27 |
| 2.5 < \( W \) ≤ 3.0 | 12 |
| 3.0 < \( W \) | 0 |
#### Analysis:
(i) **Histogram Construction and Data Distribution:**
To represent the data visually, a histogram should be created using the frequency values and corresponding weight intervals. The vertical axis will represent the number of fish (frequency), and the horizontal axis will represent the weight ranges (kg). The shape of the histogram will help in determining whether the data is symmetrical, positively skewed, or negatively skewed.
(ii) **Proposed Probability Density Functions (p.d.f.s):**
A friend of the farmer suggests modeling the weight \( W \) as a continuous random variable, proposing four possible p.d.f.s:
\[ f_{1}(w) = \frac{2}{9} w (3 - w) \]
\[ f_{2}(w) = \frac{10}{81} w^2(3 - w)^2 \]
\[ f_{3}(w) = \frac{4}{27} w^2(3 - w) \]
\[ f_{4}(w) = \frac{4}{27} w (3 - w)^2 \]
These functions are defined for \( 0 < W \leq 3 \).
(iii) **](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F732159e0-135e-441a-8a8d-d8e4bb926137%2F525c912f-ad75-436e-9200-b4ae33f7ec7c%2Fv3pcdw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Statistical Study of Trout Weights for Fish Farming Decisions
#### Problem Statement:
A fish farmer is investigating the weight distribution of trout in a lake to decide whether to net the lake and sell the fish. A sample of 100 fish was collected and weighed, yielding the following results:
#### Data:
The weights of the fish are organized in two tables, representing different weight ranges and their corresponding frequencies.
| Weight, \( W \) (kg) | Frequency |
|----------------------|-----------|
| 0 < \( W \) ≤ 0.5 | 2 |
| 0.5 < \( W \) ≤ 1.0 | 10 |
| 1.0 < \( W \) ≤ 1.5 | 23 |
| 1.5 < \( W \) ≤ 2.0 | 26 |
| Weight, \( W \) (kg) | Frequency |
|----------------------|-----------|
| 2.0 < \( W \) ≤ 2.5 | 27 |
| 2.5 < \( W \) ≤ 3.0 | 12 |
| 3.0 < \( W \) | 0 |
#### Analysis:
(i) **Histogram Construction and Data Distribution:**
To represent the data visually, a histogram should be created using the frequency values and corresponding weight intervals. The vertical axis will represent the number of fish (frequency), and the horizontal axis will represent the weight ranges (kg). The shape of the histogram will help in determining whether the data is symmetrical, positively skewed, or negatively skewed.
(ii) **Proposed Probability Density Functions (p.d.f.s):**
A friend of the farmer suggests modeling the weight \( W \) as a continuous random variable, proposing four possible p.d.f.s:
\[ f_{1}(w) = \frac{2}{9} w (3 - w) \]
\[ f_{2}(w) = \frac{10}{81} w^2(3 - w)^2 \]
\[ f_{3}(w) = \frac{4}{27} w^2(3 - w) \]
\[ f_{4}(w) = \frac{4}{27} w (3 - w)^2 \]
These functions are defined for \( 0 < W \leq 3 \).
(iii) **
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