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 trials (all independent and of probability ) will happen precisely at the -th trial.

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

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 at the bottom. Moreover, we want to have just one success, so all other trials must be unsuccessful, which explains the .

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