Definition
CDF
The Cumulative Distribution Function \(F_X(x)\) gives the probability that \(X\) is at most \(x\): $$ F_X(x) = P(X \leq x) $$
Interactive: PMF to CDF Converter
Watch how the CDF accumulates the PMF values. The CDF at any point is the sum of all PMF values up to and including that point.
Example: Die Roll
PMF: P(X = k)
CDF: P(X ≤ k)
Click on the charts to see PMF and CDF values.
Interactive: CDF Step Function
For discrete random variables, the CDF is a step function. It jumps at each value where the PMF is non-zero.
Click on a point to see F(x) at that location.
F(3) = P(X ≤ 3) = 0.5
Interval Probability: P(a < X ≤ b)
The probability that \(X\) falls in an interval can be computed using the CDF: $$ P(a < X \leq b)=F_X(b) - F_X(a) $$
Interactive: Interval Calculator
Properties of CDF
Key Properties
- ✓ Non-decreasing: If \(a < b\), then \(F(a) \leq F(b)\)
- ✓ Right-continuous: \(\lim_{x \to a^+} F(x) = F(a)\)
- ✓ Limits: \(\lim_{x \to -\infty} F(x) = 0\), \(\lim_{x \to \infty} F(x) = 1\)
Discrete vs Continuous
- Discrete: Step function with jumps
- Continuous: Smooth, continuous curve
- Mixed: Combination of both