**Given:**Let $ X \sim F_{m,n} $.**Task:** 1. Show that \[ F_{\alpha; m,n} = \frac{1}{F_{1-\alpha; n,m}} \]2. Show that \[ t^2_{1-\frac{\alpha}{2}; n} = F_{1-\alpha; 1,n} \]**Additionally:** Draw the relevant graphs for both questions.**Explanation:**The task involves proving mathematical relationships between F-distributions and T-distributions, and then illustrating these concepts with appropriate graphs. The given equations are related to statistical distributions and involve probability and critical values. For graphs, consider visualizing the properties of F-distributions and T-distributions. These graphs might show probability density functions (PDFs) highlighting critical values ($ F_{\alpha} $, $ t_{1-\frac{\alpha}{2}} $) and their relationships.

Answered: Let X Fm,n. Show that Fa;m, n…

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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Question

**Given:**

Let $ X \sim F_{m,n} $.

**Task:**

1. Show that
\[
F_{\alpha; m,n} = \frac{1}{F_{1-\alpha; n,m}}
\]

2. Show that
\[
t^2_{1-\frac{\alpha}{2}; n} = F_{1-\alpha; 1,n}
\]

**Additionally:** Draw the relevant graphs for both questions.

**Explanation:**

The task involves proving mathematical relationships between F-distributions and T-distributions, and then illustrating these concepts with appropriate graphs. The given equations are related to statistical distributions and involve probability and critical values.

For graphs, consider visualizing the properties of F-distributions and T-distributions. These graphs might show probability density functions (PDFs) highlighting critical values ($ F_{\alpha} $, $ t_{1-\frac{\alpha}{2}} $) and their relationships.

Expert Solution

Step 1: we define the given information first the we show that one by one

Given,

$X tilde F subscript m comma n end subscript$

$f left parenthesis F right parenthesis equals fraction numerator open square brackets m over n close square brackets over denominator B open parentheses m over 2 comma n over 2 close parentheses end fraction open square brackets m over n F close square brackets to the power of m over 2 minus 1 end exponent over open square brackets 1 plus m over n F close square brackets to the power of fraction numerator m plus n over denominator 2 end fraction end exponent space space comma 0 less or equal than F less or equal than infinity$

$I f space F subscript alpha semicolon m comma n end subscript space i s space a space p o i n t space s u c h space t h a t space P left square bracket space F subscript left parenthesis m comma n right parenthesis end subscript greater or equal than F subscript left parenthesis alpha semicolon m comma n right parenthesis end subscript right square bracket equals alpha space space space l. e P open square brackets space 1 over F subscript left parenthesis m comma n right parenthesis end subscript less or equal than 1 over F subscript left parenthesis alpha semicolon m comma n right parenthesis end subscript close square brackets equals alpha space a l s o P open square brackets space F subscript left parenthesis n comma m right parenthesis end subscript less or equal than 1 over F subscript left parenthesis alpha semicolon m comma n right parenthesis end subscript close square brackets equals 1 minus alpha space space space space space.............. left parenthesis 1 right parenthesis H o w space e v e r space b y space d e f i n a t i o n P open square brackets space F subscript left parenthesis n comma m right parenthesis end subscript greater or equal than F subscript left parenthesis 1 minus alpha semicolon n comma m right parenthesis end subscript close square brackets equals 1 minus alpha space space space space space............... left parenthesis 2 right parenthesis F r o m space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis space w e space g e t 1 over F subscript left parenthesis alpha semicolon m comma n right parenthesis end subscript equals F subscript left parenthesis 1 minus alpha semicolon n comma m right parenthesis end subscript F subscript left parenthesis alpha semicolon m comma n right parenthesis end subscript equals 1 over F subscript left parenthesis 1 minus alpha semicolon n comma m right parenthesis end subscript$

Hence the prove

$F subscript left parenthesis alpha semicolon m comma n right parenthesis end subscript equals 1 over F subscript left parenthesis 1 minus alpha semicolon n comma m right parenthesis end subscript$

Graph is

1import numpy as np
2import matplotlib.pyplot as plt
3
4# Define parameters
5alpha = np.linspace(0.01, 0.99, 100)  # Generate alpha values
6m = 5  # Choose arbitrary values for m and n
7n = 7
8
9# Calculate F_alpha and F_1-alpha
10F_alpha = 1 / (1 + (m / n) * alpha / (1 - alpha))
11F_1_alpha = 1 / (1 + (n / m) * (1 - alpha) / alpha)
12
13# Plot
14plt.figure(figsize=(10, 6))
15plt.plot(alpha, F_alpha, label=r'$F_{\alpha,m,n}

output is

, color='blue')
16plt.plot(alpha, F_1_alpha, label=r'$\frac{1}{{F_{1-\alpha,n,m}}}

output is

, color='red', linestyle='dashed')
17plt.title('Relationship between F-distributions')
18plt.xlabel(r'$\alpha

output is

)
19plt.ylabel('Value')
20plt.legend()
21plt.grid(True)
22plt.show()

output is

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Let X Fm,n. Show that Fa;m, n questions. = F1-ain,m and ti-n = F1-a;1,n and draw the relevant graphs for both