Metaphysics on geometric distribution in probability theory

I realized geometric distribution is not exactly about the time needed to get the first success in a given number of trials. This is a very odd feeling. It is probably a feeling applied mathematicians get sometimes, when they feel they are doing the best they can, and yet the theory is not perfect.

This may be a naive post, I warn you, but I was really stunned when I realized this.

Geometric distribution is not about the first success

Let’s jump to the point. We know (or at least, I was taught) that geometric distribution is used to calculate the probability that the first success in k trials (all independent and of probability p) will happen precisely at the k-th trial.

Remember that a geometric distribution is a random variable X such that its distribution is

\Pr(X=k)=(1-p)^{k-1}\,p\,

How can we relate the above distribution with the fact that it matches the first success? Well, we need to have one success, which explains the p at the bottom. Moreover, we want to have just one success, so all other trials must be unsuccessful, which explains the (1-p)^{k-1}.

But hey, where would first ever be written? Continue reading “Metaphysics on geometric distribution in probability theory”