The image describes different types of random variables (r.v.) based on their support $X(\Omega)$, which is the set of values the random variable $X$ can take.

Explanation:

1. Discrete Random Variable (d.r.v.):

   • If the support $X(\Omega)$ is finite or countably infinite (i.e., you can list its elements), the random variable $X$ is called discrete.
   • Example: Consider a dice roll where the possible outcomes are $\{1, 2, 3, 4, 5, 6\}$. This is a finite set, so the random variable for this outcome is discrete.

2. Continuous Random Variable (c.r.v.):

   • If the support $X(\Omega)$ is uncountably infinite (for instance, covering a continuous range of values like all real numbers within an interval), the random variable $X$ is called continuous.
   • Example: The amount of time (in hours) until a lightbulb burns out might be modeled as a continuous random variable, say between 0 and infinity, since it can take any real value in that range.

3. Absolutely Continuous Random Variable:

   • A continuous random variable is said to be absolutely continuous if its distribution function is both continuous and has a derivative (i.e., a probability density function, or PDF) almost everywhere (with some exceptions at specific points).
   • Example: The height of a population could be modeled by a normal distribution, where the PDF is a smooth curve. This makes it absolutely continuous, as the distribution has a well-defined derivative at almost every point.

Example for Absolute Continuity:

Consider the uniform distribution on the interval $[0,1]$. The probability density function (PDF) of this distribution is constant and equal to 1 over the interval $[0,1]$, and 0 otherwise: