…where is a sigma-algebra.
In other words, a random variable is a mapping from the possible outcomes in the sample space to a measurable space (in this case ).
As such, in general, we can see a random variable as the mathematical formalisation of quantities or objects that depend on randomness.
Uppercase letters, like , denote the random variable itself. On the other hand, lowercase letters, like , denote a value that the corresponding random variable may take.
With this, we can write the range of a random variable as:
There are two types of random variables:
For both of these, we can define a density function, which helps us specify the probability of the random variable being a certain value or being in a given interval.