To talk about an introduction to econometrics, it is necessary to establish the mathematical and statistical foundations to understand it. Starting from the summation:
![Econometric summation](https://usercontent.one/wp/www.javierparra.net/wp-content/uploads/2021/04/Econometric-summation.png?media=1676887272)
- The symbol Σ is the Greek capital letter sigma and means “the sum of”.
- The letter i is called the summation index; this letter is arbitrary and can also appear as t, j or k.
- The expression ∑(i=1)^n is read as the sum of Xn‘s terms from i equal to one up to n.
- The numbers i and n are the lower bound and upper bound of the summation.
Rules of the addition operation
![sum operation rules](https://usercontent.one/wp/www.javierparra.net/wp-content/uploads/2021/04/Sum-operation-rules.png?media=1676887272)
- A random variable is a variable whose value is unknown until it is observed.
- A discrete random variable can only take a limited or countable number of values.
- A continuous random variable can take any value in an interval.
- The population is the set of individuals who have certain characteristics and are of interest to a researcher.
- The sample is a subset of the population.
Population and sample moments
The sample mean, mean or expected values:
![Population and sample moments - Sample mean - Sample mean or expected values](https://usercontent.one/wp/www.javierparra.net/wp-content/uploads/2021/04/Population-and-sample-moments-Sample-mean-Sample-mean-or-expected-values.png?media=1676887272)
A monotonic relationship is one of the following:
- When the value of one variable increases, so does the value of the other.
- When the value of one variable decreases, the value of the other variable decreases.
Let’s look at the following examples of monotonic and non-monotonic relationships:
![example of monotonic and non-monotonic relationships](https://usercontent.one/wp/www.javierparra.net/wp-content/uploads/2021/04/example-of-monotonic-and-non-monotonic-relationships.png?media=1676887272)
![P Spearman](https://usercontent.one/wp/www.javierparra.net/wp-content/uploads/2021/04/P-Spearman.png?media=1676887272)
Expected values of the functions of a random variable
![Expected values of the functions of a random variable](https://usercontent.one/wp/www.javierparra.net/wp-content/uploads/2021/04/Expected-values-of-the-functions-of-a-random-variable.png?media=1676887272)
![Expected values of several random variables](https://usercontent.one/wp/www.javierparra.net/wp-content/uploads/2021/04/Valores-esperados-de-varias-variables-aleatorias.png?media=1676887272)
Some important probability distributions
Normal or Gaussian; if X is a normally distributed random variable with mean 𝝻 and variance σ2, it can be symbolised as X~N(𝝻,σ2)
![Normal or Gaussian](https://usercontent.one/wp/www.javierparra.net/wp-content/uploads/2021/04/Normal-or-Gaussian.png?media=1676887272)
Chi-square: If X, is a normally distributed random variable with mean 0 and variance σ2, then V= X12 +X22 +…+Xm2 ~ X(m)2
![Chi-square](https://usercontent.one/wp/www.javierparra.net/wp-content/uploads/2021/04/Chi-square.png?media=1676887272)
A Student’s random variable is formed by dividing a standard normal random variable with mean 0 and variance 1 by the square root of an independent chi-squared random variable, V, divided by its m degrees of freedom.
![Student's random variable](https://usercontent.one/wp/www.javierparra.net/wp-content/uploads/2021/04/Students-random-variable.png?media=1676887272)