Why is geom_density plotting my data differently from the expected image? - r

Question

I have the following task statement:
In this task we want to simulate random variables with density
To do this, write a function r_density(n) that simulates n of such random variables.
Then use this function to simulate N = 1000 of such random variables. Using geom_density() you can now estimate the density from the simulated random variables. We can compare this estimate with the real density. To do this, create a graph that looks like this:
Problem is, however, that I don't understand why my output looks like this:
Why is the raked density plotted in such a weird way? Can someone explain to me why it looks like that and how to get the estimated density from the expected image?
This is the corresponding code I wrote for the above plot:
library(tidyverse)
N <- 1000
r_density <- function(n){
exp(-abs(n))/2
}
x <- runif(N)
tb <- tibble(
x = x,
density_fkt = r_density(x)
)
ggplot() +
geom_density(
data = tb,
mapping = aes(
x = density_fkt,
y = ..scaled..
)
) +
geom_function(
fun = r_density,
xlim = c(-6,6),
color = "red",
size = 1
) +
theme_minimal() +
labs(
x = "x",
y = "Dichtefunktion f(x)",
title = "Geschätzte (schwarz) vs echte (rot) Dichte"
)

Answer 1

You may use inverse transform sampling or rejection sampling. I choose rejection sampling.
library(tidyverse)
N <- 1000
r_density <- function(n){
exp(-abs(n))/2
}
x = c()
while (length(x) < N) {
y = rnorm(1)
while (y > 6 | y < -6) {
y = rnorm(1)
}
u = runif(1)
if (u < r_density(y)/(dnorm(y) * 3)) {
x=append(x, y)
}
}
tb <- tibble(
x = x,
density_fkt = r_density(x)
)
ggplot() +
geom_density(
data = tb,
mapping = aes(
x = x,
y = ..density..
)
) +
geom_function(
fun = r_density,
xlim = c(-6,6),
color = "red",
size = 1
) +
theme_minimal() +
labs(
x = "x",
y = "Dichtefunktion f(x)",
title = "Geschätzte (schwarz) vs echte (rot) Dichte"
)

Answer 2

Here's the inverse transform sampling method (this involves some difficult integration, so perhaps not what your teacher intended)
r_density <- function(n) {
cdf <- function(x) {
1/4 * exp(-x) * (-1 + 2 * exp(x) + exp(2*x) - (-1 + exp(x))^2 * sign(x))
}
sapply(runif(n), function(i) {
uniroot(function(x) cdf(x) - i, c(-30, 20))$root
})
}
Plotting gives:
ggplot() +
geom_density(aes(r_density(1000))) +
geom_function(
fun = function(x) exp(-abs(x))/2,
xlim = c(-6,6),
color = "red",
size = 1
) +
theme_minimal() +
labs(
x = "x",
y = "Dichtefunktion f(x)",
title = "Geschätzte (schwarz) vs echte (rot) Dichte"
)