Order statistics
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Week 2, Math 563, Sp21
Background
What is the purpose of this worksheet?
- Learn using basic R, practice using Markdown;
- Create simulations for order statistics.
For more info, formatting, and technical background, see previous handout (on sampling distributions).
Installing and loading packages
Just like every other programming language you may be familiar with, R’s capabilities can be greatly extended by installing additional “packages” and “libraries”.
To install a package, use the install.packages() command. You’ll want to run the following commands to get the necessary packages for today’s lab:
install.packages("ggplot2")You only need to install packages once. Once they’re installed, you may use them by loading the libraries using the library() command. For today’s lab, you’ll want to run the following code
library(ggplot2) # graphics libraryProblem 1: \(N(0,1)\)
- Let us pick sample size \(n=10\). Let \(X\sim N(0,1)\).
- Repeat the following \(10,000\) times:
- Draw a random sample \(X_1,\dots,X_n\) from the normal density \(f(x)\).
- For \(k\in\{1,3,5,9,10\}\), compute the \(k\)-th statistic \(y_k\).
- Draw a random sample \(X_1,\dots,X_n\) from the normal density \(f(x)\).
- Plot the histogram of \(y_k\) for each \(k\).
# your code hereDiscuss. What do you see?
Problem 2: \(Gamma(1.5,5)\)
- Let us pick sample size \(n=25\). Let \(X\sim Gamma(1.5,5)\).
- Repeat the following \(10,000\) times:
- Draw a random sample \(X_1,\dots,X_n\) from the normal density \(f(x)\).
- For \(k\in\{2,4,16,24\}\) compute the \(k\)-th statistic \(y_k\).
- Draw a random sample \(X_1,\dots,X_n\) from the normal density \(f(x)\).
- Plot the histogram of \(y_k\) for each \(k\).
# your code hereDiscuss. What do you see?
License
This document is created for Math 563, Spring 2021, at Illinois Tech, taught by Sonja Petrovic1
While the course materials are generally not to be distributed outside the course without permission of the instructor, this particular set of notes is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
