Poisson Distribution

set.seed(42)  ##Meaning of Life

u<- replicate(100, mean(rpois(n=30,lambda=40)))
hist(u)

summary(u)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   36.43   39.22   40.13   39.98   40.67   42.30
min <- replicate(100, min(rpois(n=30,lambda=40)))

hist(min)

summary(min)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   20.00   25.00   28.00   27.44   30.00   33.00

The means are normally distributed with center at lambda=40 and standard deviation equal the the standard error. if you increase the number of samples you see a strong trend towards normaility.

The mins are poison distributed

set.seed(42)  ##Meaning of Life

u<- replicate(10000, mean(rpois(n=30,lambda=40)))
hist(u)

summary(u)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   35.43   39.20   40.00   39.99   40.77   44.50
min <- replicate(10000, min(rpois(n=30,lambda=40)))

hist(min)

summary(min)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   16.00   26.00   28.00   27.66   30.00   36.00