## Time Series: why do we need Stationarity and Ergodicity

A time series is a series of data points indexed in time order, normally with equally spaced points in time. Examples of time series are stocks’ prices, monthly returns, company’s sales and so forth. Time series can be seen as data with a target variable (price, returns, amount of sales…) and one feature only: time. …

## Maximum Likelihood Estimation

The main goal of statistical inference is learning from data. However, data we want to learn from are not always available/easy to handle. Imagine we want to know the average income of American women: it might be unfeasible or highly expensive to collect all American women’s income data. Besides, even in a scenario when this …

## Markov Chain Montecarlo

A Markov chain can be defined as a stochastic process Y in which the value at each point at time t depends only on the value at time t-1. It means that the probability for our stochastic process to have state x at time t, given all its past states, is equal to the probability …

## Understanding Rejection Sampling method

Rejection sampling is a computational technique whose aim is generating random numbers from a target probability distribution f(x). It is related to the general field of MonteCarlo methods, whose core is generating repeated random sampling to make numerical estimation of unknown parameters. Some words about Randomness One might ask why a random variable with probability …

## Combinatorics: permutations, combinations and dispositions

Combinatorics is that field of mathematics primarily concerned with counting elements from one or more sets. It can help us counting the number of orders in which something can happen. In this article, I’m going to dwell on three different types of techniques: permutationsdispositionscombinations Permutations Those are the easiest to compute. Imagine we have n objects, different among each others. …

## One-way Analysis of Variance (ANOVA) with Python

When you are dealing with data which are presented to you in different groups or sub-populations, you might be interested in knowing whether they arise from the same population, or they represent different populations (with different parameters). Let’s consider the following picture: As you can see, there are three different footpaths. Now the question is: …

## Hypothesis tests with Python

In my previous article, I’ve been talking about statistical Hypothesis tests. Those are pivotal in Statistics and Data Science since we are always asked to ‘summarize’ the huge amount of data we want to analyze in samples. Once provided with samples, which can be arranged with different techniques, like Bootstrap sampling, the general purpose is …

## Understanding Geometric and Inverse Binomial distribution

In my previous article, I’ve been talking about two of the most popular probability distributions of discrete random variables: Bernoulli and Binomial. Here, I’m going dwell on their so-called ‘counterparts’, which are Geometric and Inverse Binomial. Both of them concerns the idea of a sequence of Bernoulli trials, hence it is worth it to recall …

## Convergence of Random Variable

When we talk about convergence of random variable, we want to study the behavior of a sequence of random variables {Xn}=X1, X2,…,Xn,… when n tends towards infinite. Basically, we want to give a meaning to the writing: A sequence of random variables, generally speaking, can converge to either another random variable or a constant. However, …

## Conditional Probability and Rare Events

Conditional probability refers to the probability of a generic event, given some extra information. More specifically, the conditional probability of one event A with respect to B: Expresses the probability of A given that B has occurred. If the two events are independent, the simple and conditional probability coincides (the occurrence of B has nothing …