Cumulative Distribution Function (CDF)
What is CDF?
A Cumulative Distribution Function (CDF) of a continuous random variable \(X\) gives the total probability accumulated up to and including a specific value \(x\). It answers:
“What is the probability that \(X \leq x\)?”
Interpretation
- \(F_X(x)\) is the area under the PDF curve from \(-\infty\) to \(x\).
- It shows how much probability has accumulated up to that point.
- The CDF is non-decreasing and always lies in the range \([0, 1]\).
- For small values \(x\), \(F_X(x) \approx 0\); for large values, \(F_X(x) \approx 1\).
Properties
| Property | Description |
|---|---|
| Monotonicity | \(F_X(x_1) \leq F_X(x_2)\) for \(x_1 < x_2\) |
| Limits | \(\lim_{x \to -\infty} F_X(x) = 0\), \(\lim_{x \to \infty} F_X(x) = 1\) |
| Range | \(0 \leq F_X(x) \leq 1\) |
| Continuity | \(F_X(x)\) is continuous from the right |
Relationship to PMF/PDF
The CDF and PMF/PDF are tightly connected:
- To get the PMF:
\[
p_X(x) = F_X(x) - F_X(x^{-}) = \mathbb{P}(X = x)
\]
- To get the PDF, differentiate the CDF:
\[
f_X(x) = \frac{d}{dx} F_X(x)
\]
Example
Let \(X \sim \mathcal{N}(165, 10^2)\) be a normal distribution modeling women’s heights with:
- Mean \(\mu = 165\)
- Standard deviation \(\sigma = 10\)
Then:
- \(F_X(160) \approx 0.31\): 31% of women are shorter than 160 cm.
- \(F_X(175) \approx 0.84\): 84% are shorter than 175 cm.
The CDF increases smoothly from 0 to 1, reflecting accumulated probability.