6.1

Running Code

library(tidyverse)

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library(purrrfect)

Attaching package: 'purrrfect'

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

    replicate, tabulate

lambdas <- c(0.5, 1, 1.5, 2)
n_sims <- 10000

sim_df <- data.frame(lambda = numeric(0), U = numeric(0))

for(lam in lambdas){
  X <- rexp(n_sims, rate = lam)
  Y <- rexp(n_sims, rate = lam)
  U <- X + Y
  sim_df <- rbind(sim_df, data.frame(lambda = rep(lam, n_sims), U = U))
}
  
  u_grid <- seq(0, max(sim_df$U), length.out = 1000)
  analytic_df <- map_dfr(lambdas, function(lam){
    data.frame(lambda = lam,
               U = u_grid,
               density = dgamma(u_grid, shape = 2, rate = lam))
  })
ggplot(sim_df, aes(x = U))+
  geom_histogram(aes(y = after_stat(density)),
            binwidth = 0.1, fill = 'cornflowerblue', color = 'black')+
  geom_line(data = analytic_df, aes(x = U, y = density), color = "red",size = 1 )+
  facet_wrap(~lambda, scales = 'free')+
  labs(title = "Simulated vs Analytic Density of U = X + Y",
       x = "U", y = "Density")+
  theme_classic()

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