202405171700
Status: #idea
Tags: Probability, Statistics

Random Variable

In the formal mathematical language of measure theory, a random variable is defined as a measurable function from a probability measure space (called the sample space) to a measurable space.

Indeed a random variable is a measure map of one dimension where the space being mapped to is the real line $\mathbb{R}$.

Continuous Random Variables

$X$ is a continuous random variable if $F$ (the Cumulative Density Function of $X$) is continuous.
Observe that $f(x)$ (if it exists) may not be continuous, this does NOT mean that the random variable is also not continuous.

Note that if $f(x)$ is continuous, then it's cumulative density function (and therefore the associated random variable) will be continuous.

In general to check if a Cumulative Density Function is continuous from the probability density function, you integrate the function and then check if there's any discontinuity (weird jumps) in the functions' path (just graph it).

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