About This Assignment

This mini project covers chapter 6 of MS in a more in-depth way than the previous quiz. You will be tasked with answering a challenging word problem. You will be asked to show your work on the next assignment page by submitting HTML.

Format: numeric answer "quiz"
Time limit: none
Number of attempts: 3

Academic Integrity

Getting help from your peers is not permitted.
Collaborating, or completing the quiz with others in the class, is not permitted.
Copying or reusing previous work is not permitted.
Open-book research for the quiz is permitted and encouraged.

Introduction

There are some very important statistical applications where you will need to use order statistic theory to find the probability density functions of relevant sampling statistics. The theory related to some of the random variables is clear while others are more difficult to find, especially closed-form algebraic expressions for the density. Monte Carlo Simulation theory can be made to do service in such cases and even in cases where a density is algebraically tractable it may not be known well -- in which case estimates from the simulation will still be of great use.

Calculation

Suppose $LaTeX: Y_i \stackrel{iid}{\sim}Unif(0,M)$ and a sample of size n = 10 is made with M = 200:

What is the algebraic expression for the pdf of Y_max ?
What is the algebraic expression for the pdf of Y_min ?
Edit the simulate Y max file (R) to answer the following:
1. Simulate the sampling distribution of Y_max and add the curve of the pdf from 1. onto the histogram (Must use freq = FALSE in hist())
2. Simulate the sampling distribution of Y_min and add the curve of the pdf from 2. onto a different histogram (Must use freq = FALSE in hist())
3. Use iter = 1000, 10000,100000 for each of the two simulations
Create an RMD document and express your final pdf's using Latex - all derivation can be done with a picture.
Make your R function create all 6 plots, knit to HTML.
Answer the quiz question

Example simulation

Note the histogram and the density curve

Histogram: Density on Y axis, simulated maximum value of the sample on X axis