Recap of special probability distributions(1)

This and the following log will be a two part recap of special probability distributions. We will go over the formulation of the distributions and some typical modeling cases together.

Below is a mind map of the special probability distributions from Oliver Ibe’s Fundamentals of Applied Probability and Random Processes.

For part 1, we will be mainly focusing on discrete random variables.

Bernoulli distribution

Binomial distribution

Geometric distribution

Modified geometric distribution

Finding counts until first failure instead; Just exchange p & (1-p) in the formulation.

The derivative trick for sums

When computing the expectation and variance of the geometric distribution, we can construct a derivative form. This form allows us to transform the computation from ‘sum of derivatives’ to ‘derivatives of sum’. In this case, it helps us the address the infinite sum and simplifies the calculation greatly. Below is the example:

Computation of the first moment E[X] =>

Computation of the second moment E[X²] =>

The memoryless property

If we know that there were no success in the previous n attempts, the probability for the next k attempt from n to be the first successful one is the same as starting from next k attempt from zero. (Note that this the memoryless property may be reflect cases in real life correctly, for instance, a series of earthquakes is highly likely to be related):

Here’s a graph illustrating the memoryless of exponential distribution (指数分布「Exponential Distribution」 – 知乎), which offers a geometric approach to memoryless property – considering the similarity of shapes.

Pascal-k Distribution

Poisson distribution

For modeling the occurrence (k) of an event within a period of time. The parameter lambda

The value of both expectation and variance is lambda.

A trick of memorization: The Taylor series of e^x (Since you are summing for to check the normalization condition)

Binomial to Poisson to Exponential