202405301238
Status: #idea
Tags: Probability Theory
State: #nascient

Cumulative Density Function

The definition of a cumulative density function often shortened cdf, is as follows:

So we see the cumulative nature, what is the probability that $X$ is at value $x$ or anything below that. Therefore it is non-decreasing.
It is also sometimes represented as

All of these notations are equivalent, but most of the time the first one is used.

It is strongly related to Probability Density Functions or Probability Mass Function and their associated Random Variable, but be careful rigorously a cumulative density function is not defined as the integral of a pdf or any such nonsense, in fact quite often we will have a cumulative density function without a pdf even existing. As long as you have a probability space, which we always do (lel). All you need to do to compute the cdf is to feed the set $(-\mathrm{\infty },x]\cap \Omega$ to your probability measure $P$)

Because of how they function, if you want to compute $P(X\le x)$, you can simply plug $x$ in your $F$.

But you're given a cdf, and you want to compute $P(a\le X\le b)$, you're going to need to remove the cumulative probability up to $a$ from the cumulative up to $b$. In this case we'd want to keep $a$ so we'd actually go $a^{-}$, in a continuous context this makes no difference, but in a discrete context this means decreasing by one.

Continuous Cumulative CDF

The CDF is continuous

Discrete Cumulative CDF

The CDF is a step-function, in other words $F^{\prime }(x)=0$ except at jump points where $F^{\prime }(x)$ does not exist.

The Rest CDF

The CDF is not continuous, but is not a step function either.

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