sds240_pset09_f23

S&DS 240 Problem Set 9, Fall 2023 Assignment info • Due date: November 10, 11:59PM in Gradescope. Be sure to assign pages to the problems when you submit. If you do not, there will be an 𝑛 -point penalty per problem where 𝑛 is the problem set number. • You need to submit your solutions as a pdf file. This can be in any of the forms – handwritten scanned or photographed – L A T E X – knitted Rmarkdown or Quarto – MS Word exported to pdf • Your work should be legible and clear. Use complete sentences and explain your approach and thought processes. If your work is messy or unclear, you won’t receive credit. • Start the assignment early and work through the problems in a separate “rough draft” notebook. If you get stuck, don’t panic! Obstacles and resistance of this sort is a necessary part of the learning process. When you come back to the problem later, you’ll be amazed at how often you are able to make progress. • Utilize the offce hours of the instructor, TF, and ULA. We are here to help and support you, and want to see you succeed. • Your lowest homework grade will be dropped. • You are encouraged to work together, however you must write your solutions yourself. The last question asks you to credit any classmates you collaborated with, and other sources you used. 1

Question 1: Chapter 5, Exercise 1 Work saving tip: first find the CDF of 𝑋 and use it to answer the questions. 2

Question 3: Chapter 5, Exercise 13 4

Question 4: Chapter 5, Exercise 23 For part (a) give your answer in terms of Φ (the CDF of the standard normal) 5

Question 6: Chapter 5, Exercise 18 Replace part (b) with the following: Using a Unif (0, 1) distribution generate 10,000 random numbers from Pareto distributions with parameters 𝑎 = 1/2 , 𝑎 = 2 , and 𝑎 = 10 . Plot histograms of each next to each other in a single row. Include the parameter value in the title of each histogram. For examples of R code to generate random numbers from a desired distribution using a uniform, see the continuous_rvs.html file in Canvas. Here is R code to plot three histograms next to each other using made up data. Give the histograms appropriate titles (the main argument), and feel free to experiment withe col argument to choose any color(s) that you like ( Yale blue is “#00356B”). You may leave the 𝑥 -axis labels blank. # generate three sets of Unif(0, 1) random numbers u1 <- runif ( 1e4 ) u2 <- runif ( 1e4 ) u3 <- runif ( 1e4 ) # these are made up data, you'll be using a different formula based on part (a) x1 <- log (u1) x2 <- sin (u2) x3 <- u3 # Plot histograms: # the first command sets up the plot window to show # three plots next to each other in one row par ( mfrow = c ( 1 , 3 )) hist (x1, xlab = "" , main = "Plot title 1" , col = "blue" ) hist (x2, xlab = "" , main = "Plot title 2" , col = "green" ) hist (x3, xlab = "" , main = "Plot title 3" , col = "orange" ) Plot title 1 Frequency -8 -6 -4 -2 0 0 1000 2000 3000 4000 Plot title 2 Frequency 0.0 0.2 0.4 0.6 0.8 0 200 400 600 Plot title 3 Frequency 0.0 0.2 0.4 0.6 0.8 1.0 0 100 200 300 400 500 par ( mfrow = c ( 1 , 1 )) # this command resets the plot window to showing a single plot at a time 7

Question 7: Indicate any students you collaborated with and any other sources you used (you do not need to indicate the textbook or course notes). If you did not use any sources (other than the textbook and lecture notes), please write “No other sources used”. 8