Cumulative Distribution Function

A cumulative distribution function is defined as the sum of the probabilty from the left to some point \(x\) on the right,

\[ F(x) = \int_{-\infty}^x f_X~\mathrm dx \]

• Non-decreasing

• Right continuous

• Decreasing to zero $\( \lim_{x\rightarrow -\infty} F(x) = 0 \)$

• Increasing to one $\( \lim_{x\rightarrow \infty} F(x) = 1 \)$

CDF to PDF

The CDF is the anti-derivative of the PDF,

\[ f_X(x) = \frac{\mathrm d F}{dx} \]

Proof : Recall that the anti-derivative is defined by an indefinite integral. The first axiom of probabilty implies that \(f_X\) converges from towards the left \(f(-\infty) \rightarrow 0\). Using this fact, the indefinite integral of \(f_X(x)\) can be described by the left bound of \(-\infty\) and an arbritrary right bound \(x\).

$$ \int f_X(x)~\mathrm dx = \int_{-\infty}^{x} f_X(x)~\mathrm dx = F(x)$$