23. Write a program that picks a random number between 0 and 1 and computes the negative of its logarithm. Repeat this process a large number of times and plot a bar graph to give the number of times that the outcome falls in each interval of length 0.1 in [0, 10]. On this bar graph plot a graph of the density \(f(x) = e^{−x}\). How well does this density fit your graph?

First, let’s set our seed for reproducability

set.seed(12345)

Next, let’s generate a large number of random numbers. For our purposes, let’s do 10,000

# Number of trials
n <- 1000

x = c(seq(n))
y <- c()

for (i in 1:n){
  # Grab one random negativer log from unifrom distribution between 0 and 1 and store it in y vector
  result <- -log(runif(n=1, min=0, max=1))
  y <- append(y, result)
}

NOw let’s plot our results and the function \(F(x) = e^{-x}\).

# Put x and y vectors into dataframe
df <- data.frame(x, y)

log_func <- function(x){-log(x)}

# plot histogram binned by each interval of length 0.1
ggplot(df, aes(y)) + geom_histogram(binwidth=0.1, boundary=0.1) +
  stat_function(fun=log_func, colour='red') + xlab("x")

The density function above (plotted in red) doesn’t fit our distribution of randomly-generated perfectly, but does reflect a similar exponential shape. This scaling comes in with a coefficient that ensures the density function integrates to 1, and we can use it to reflect probailities, as a probability over 1 doesn’t make sense.