The link between Poisson and Exponential distribution

When you are dealing with random experiments, linked to a set of possible outcomes, it is useful to assign to each of the possible outcomes (which might be not numerical, like events) a real number, so that you can make useful computations.

The function which assigns to each event of the space of outcomes a real number is called Random Variable. More formally, given a measurable space ( Ω, A), a random variable is a function defined on Ω with codomain in R, Borel-measurable with respect to the algebra A.

A Random variable can be:

• Discrete, if it can only take a countable number of values
• Continuous, if it can take infinitely many values

Random variables are provided with a probability distribution function, which assigns to each value of the function X a number between 0 and 1. This number is a probability and it defines the likelihood of occurrence of that outcome. We will refer to that function as p(x). Note that, in order to be a probability function, the function p(x) has to respect two conditions:

• p(x) cannot be negative (non-negativity)
• the summation of all of its values (in the continuous case, the integral between minus infinity and infinity) must be equal to 1 (normalization condition)