Notation for features (general notation for continuous and discrete random variables)

2

I'm looking for the right notation for features from different types. Let us say that my samples as $m$ features that can be modeled with $X_1,...,X_m$. The features Don't share the same distribution (i.e. some categorical, some numerical, etc.). Therefore, while $X_i$ might be a continuous random variable, $X_j$ could be a discrete random variable.

Now, given a data sample $x=(x_1,...,x_m)$, I want to talk about the probability, for example, $P(X_k=x_k)<c$. But $X_k$ might be a continuous variable (i.e. the height of a person). Therefore, $P(X_k=x_k)$ will always be zero. However, it can also be a discrete variable (i.e. categorical feature or number of kids).

I'm looking for a notation that is equivalent to $P(X_k=x_k)$ but can work for both continuous and discrete random variables.

Yael M

Posted 2020-05-27T08:02:00.680

Reputation: 21

Answers

0

As far as I am concerned, there is no distinction between a continuous and a discrete variable when it comes to notation. So $P(X_k=x_k)$ is perfectly fine for either.

Valentin Calomme

Posted 2020-05-27T08:02:00.680

Reputation: 4 666

To my knowledge, if $X$ is a continuous variable then for each constant $c$, $P(X=c)=0$. Instead of constants, we should talk about intervals and measure the probability using a probability density function. – Yael M – 2020-05-27T11:46:43.813

0

Maybe relying on set notation would work?

$P(X_k \in s_k)$ where:

  • $s_k = \{ x_k \}$ if $X_k$ is discrete
  • $s_k = [ x_k-\epsilon , x_k+\epsilon]$ if $X_k$ is continuous

Erwan

Posted 2020-05-27T08:02:00.680

Reputation: 12 600