Cumulative Distribution Functions

The cumulative distribution function (CDF) is one of the most versatile tools in statistics. It provides a complete description of the distribution and allows you to calculate probabilities, find percentiles, and connect the PDF and CDF through differentiation and integration.

---

Definition

The cumulative distribution function of a random variable $X$X is:

$F(x) = P(X \leq x) = \int_{-\infty}^x f(t) \, dt$F(x)=P(X≤x)=∫−∞x​f(t)dt

Property	Detail
$F(-\infty) = 0$F(−∞)=0	No probability below the minimum
$F(\infty) = 1$F(∞)=1	All probability is accounted for
$F$F is non-decreasing	If $a < b$a<b, then $F(a) \leq F(b)$F(a)≤F(b)
$F$F is continuous (for continuous $X$X)	No jumps in the CDF

---

Relationship Between PDF and CDF

The PDF and CDF are linked by:

$F(x) = \int_{-\infty}^x f(t) \, dt \quad \text{and} \quad f(x) = F'(x)$F(x)=∫−∞x​f(t)dtandf(x)=F′(x)

In other words:

• Integrate the PDF to get the CDF
• Differentiate the CDF to get the PDF