Running Monte Carlo simulations

The Monte Carlo simulation (see the example, http://news.mit.edu/2010/exp-monte-carlo-0517 or https://en.wikipedia.org/wiki/Monte_Carlo_method) is one of the elementary computational techniques. In this recipe, we will explain how it can be implemented efficiently in Julia.

Getting ready

Consider the following problem. Assume that on each day, a random volume of water leaks from a pipe to a container. The amount of water that leaks out is greater than zero, but less than one. How many days do we need to wait till a container having a volume equal to one is filled, if on each day the amount of water that leaks is a uniformly random value between zero and one?

Formally, we repeatedly draw independent random numbers from a uniform distribution on the 

interval. How many draws, on average, are required until the sum of drawn numbers is greater than or equal to 1? Let

 be a sequence of independent random variables. We want to find the following:

We will approximate...