6.2

Running Code

library(tidyverse)

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── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
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✔ purrr     1.0.4     
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
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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

param_tbl <- parameters(~alpha, ~beta, 
        c(1), c(2)) %>%
  add_trials(10000)

sim_df <- param_tbl %>%
  rowwise() %>%
  mutate(
    Y = rbeta(1,alpha,beta),
    U = 1 -2*Y, 
    V = 1/Y,
    W = sqrt(Y)
  )
fY <- function(y) ifelse(y> 0 & y < 1, 2 *(1-y), 0)
fU <- function(u) ifelse(u > -1 & u < 1, (1 + U)/2, 0)
fV <- function(v) ifelse(v > 1, 2*(v-1)/v^3, 0)
fW <- function(w) ifelse(w > 0 & w < 1, 4*w*(1-w^2), 0 )

ggplot(sim_df, aes(x = Y))+ 
  geom_histogram(aes(y = after_stat(density)),
                 bins = 40, fill = "goldenrod", color = "black")+
  stat_function(fun = fY, color = "red", size = 1)+
  theme_classic(base_size = 14) + 
  labs(x = "Y", y = "Density", title = "Histogram of beta")

Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
ℹ Please use `linewidth` instead.

ggplot(sim_df, aes(x = U)) + 
  geom_histogram(aes(y = after_stat(density)), bins = 40, fill = "cornflowerblue", color = "black")+
  stat_function(fun = fU, color = "red", size = 1.2)+
  theme_classic()+
  labs(title = "Histogram of U", x="U", y = "Density")

Warning: Computation failed in `stat_function()`.
Caused by error in `fun()`:
! object 'U' not found

ggplot(sim_df, aes(x = V)) + 
  geom_histogram(aes(y = after_stat(density)), bins = 40, fill = "lightgray", color = "black")+
  stat_function(fun = fV, color = "red", size = 1.2)+
  coord_cartesian(xlim=c(1,5)) +
  theme_classic()+
  labs(title = "Histogram of V", x ="V", y = "Densisty")

ggplot(sim_df, aes(x = W)) +
  geom_histogram(aes(y = after_stat(density)), bins = 40, fill = "orange", color = "black")+ 
  stat_function(fun = fW, color = "red", size = 1.2)+
  theme_classic()+
  labs(title = "Histogram of W", x = "W", y = "Density")