The distribution of the sample max

An animation

Students often leave a first-semester statistics course thinking basically everything is normal, or at least if we sample enough, we’ll end up with a normal distribution. The animation here is the distribution of the sample maximum from an exponential rate 0.25 distribution for increasing sample sizes.

As you can see, the distribution is skewed to the right, and this skewness doesn’t go away as the sample size increases. As a quiz question, why does the distribution move to the right?

This is a short post, but hopefully illustrative. Below is the graph of the exponential rate 0.25 density, and the R code for the animation is at the end.

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R code for animation

# Libraries

library(dplyr)
library(tidyr)
library(ggplot2)
library(gganimate)


## Colors

MyPurple <- "#5B005B"
MyLightP <- "#dfdbdf" 
MyLightP2 <-  "#f8f4f8"  
MyLightP3 <- "#fcfafc" 


## Data

nReplicate <- 10000

Data <- data.frame("Sample" = replicate( nReplicate, max( rexp( 1, rate = 0.25 ))), "Number" = 1)

for (i in c(2, 4, 6, 8, 10, 15, 20, 30, 40, 50, 100)) {
  DataTemp <- data.frame( "Sample" = replicate( nReplicate, max( rexp( i, rate = 0.25))), "Number" = i)
  Data <- rbind( Data, DataTemp)
}

p <- ggplot(Data, aes( Sample, fill = factor( Number))) + 
  geom_density( alpha = 0.4, bw = 3) + 
  scale_x_continuous( limits = c(-5, 40)) +
  transition_states( Number, transition_length = 3, state_length = 1 ) +
  enter_fade() +
  exit_fade() +
  shadow_mark( alpha = alpha / 4, color = alpha( "grey", 0.25)) +
  guides( fill = "none") +
  ease_aes( "linear") +
  theme( strip.background = element_blank(), strip.text.x = element_blank())+
  theme( axis.text = element_text( size = 14),
         axis.title = element_text( size = 16),
         plot.title = element_text( size = 20),
         plot.background = element_rect( fill = MyLightP3),
         panel.background = element_rect( fill = MyLightP), 
         legend.background = element_rect( fill = MyLightP2),
         plot.caption = element_text( hjust = c(1), size = c(14),
                                      color = c(MyPurple))) +
  labs(title="Distribution of Sample Maximum from an Exponential rate 0.25 Distribution\n Sample Size = {closest_state}",
       y = NULL, x = NULL,
       caption = c("Briefed by Data || Thomas J Pfaff"))+
  theme(legend.position = "none")


animate(p, fps = 5, duration = 10, width = 1456/2, height = 936/2)