PS 6.4

library(dplyr)

Attaching package: 'dplyr'

The following objects are masked from 'package:stats':

    filter, lag

The following objects are masked from 'package:base':

    intersect, setdiff, setequal, union

library(tidyverse)

── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ forcats   1.0.0     ✔ readr     2.1.5
✔ ggplot2   3.5.2     ✔ stringr   1.5.1
✔ lubridate 1.9.4     ✔ tibble    3.3.0
✔ purrr     1.1.0     ✔ tidyr     1.3.1

── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

library(purrrfect)

Attaching package: 'purrrfect'

The following objects are masked from 'package:base':

    replicate, tabulate

sim_uv <- tibble(
  X = runif(10000, 0, 1),
  Y = runif(10000, 0, 1)
) %>%
  mutate(
    U = X - Y,
    V = X + Y
  ) 

ggplot(sim_uv, aes(x = U, y = V)) +
  geom_point(alpha = 0.2, size = 0.7) +
  theme_minimal()