── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr 1.1.4 ✔ readr 2.1.5
✔ forcats 1.0.0 ✔ stringr 1.5.1
✔ ggplot2 3.5.2 ✔ tibble 3.3.0
✔ lubridate 1.9.4 ✔ tidyr 1.3.1
✔ purrr 1.1.0
── 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
Attaching package: 'purrrfect'
The following objects are masked from 'package:base':
replicate, tabulate
N <- 10000
dY <- function(y) {
ifelse(y > 0 & y < 1, 2*(1-y), 0)
}
dU <- function(u) {
inside <- u > -1 & u < 1
y <- (1-u)/2
out <- dY(y) * 0.5
ifelse(inside, out, 0)
}
dV <- function(v) {
y <- 1 / v
ifelse(v > 1, dY(y) * (1 / v^2), 0)
}
dW <- function(w) {
inside <- w > 0 & w < 1
y <- w^2
out <- dY(y) * (2*w)
ifelse(inside, out, 0)
}
(sim_grid <- parameters(
~dummy, 1
) %>% add_trials(10000))
# A tibble: 10,000 × 2
dummy .trial
<dbl> <dbl>
1 1 1
2 1 2
3 1 3
4 1 4
5 1 5
6 1 6
7 1 7
8 1 8
9 1 9
10 1 10
# ℹ 9,990 more rows
sim_res <-
sim_grid %>%
mutate(
Y = rbeta(10000, shape1 = 1, shape2 = 2)
) %>%
mutate(
U = 1 - 2 * Y,
V = 1 / Y,
W = sqrt(Y)
) %>%
mutate(
fY = dY(Y),
fU = dU(U),
fV = dV(V),
fW = dW(W)
)
ggplot(sim_res, aes(x = U)) +
geom_histogram(aes(y = after_stat(density)),
bins = 40,
fill = "gray80", color = "black") +
geom_line(aes(y = fU),
linewidth = 1, color = "black") +
labs(title = "U = 1 - 2Y",
x = "u", y = "density") +
theme_classic()
ggplot(sim_res, aes(x = V)) +
geom_histogram(aes(y = after_stat(density)),
bins = 40,
fill = "gray80", color = "black") +
geom_line(aes(y = fV),
linewidth = 1, color = "black") +
labs(title = "V = 1 / Y",
x = "v", y = "density") +
coord_cartesian(xlim = c(1, 10)) +
theme_classic()
ggplot(sim_res, aes(x = W)) +
geom_histogram(aes(y = after_stat(density)),
bins = 40,
fill = "gray80", color = "black") +
geom_line(aes(y = fW),
linewidth = 1, color = "black") +
labs(title = "W = sqrt(Y)",
x = "w", y = "density") +
theme_classic()
