Why is the theoretical mode of exponential distribution different than numerically simulated one

Question

I calculated the theoretical mode for X ~ Expo(lambda) and got it as 0. However, when I simulated the scenario in R, I am getting some values. My code is shown below:

sample_mode <- function(x) {
  t <- table(x)
  m <- max(t)
  as.numeric(names(t[t==m]))
}
simfunc <-function(){
  x <- rexp(10000,2)
  median <- median(x)
  mode <- sample_mode(x)

  cat("The median is", median)
  cat("The mode is", mode)
}

The median matches, but I am getting a vector of values for the mode.

Answer 1

Exponential distribution is a distribution for a continuous random variable. For continuous random variables $\Pr(X=x)=0$ for any $x$, so for your variable you would expect each of the distinct values not more then once in your sample, so your function for calculating mode would not produce any reasonable result. Here you can find some hints how you could calculate mode for continuous variable.