Cumulative Distribution Function (CDF)

Deep Breakdown

What it tells me (Situation Awareness): The CDF for a continuous RV is the integral of the PDF from $-\infty$ to $x$. It gives the probability that $X$ takes a value at or below $x$. It is a smooth, non-decreasing function from 0 to 1. Used for percentile computation, the inverse CDF (quantile function), probability of intervals, and the probability integral transform.

Flowchart

Real-world use

Where it is used

Percentile/quantile computations (median, IQR). P-value computation in hypothesis tests. Generating random samples via inverse CDF transform. Probability integral transform (PIT) for residual diagnostics.

Problem it solves

Converts from density (PDF) to probability (area under curve), enabling interval and tail probability queries.

How it solves the problem

Integrates the PDF from $-\infty$ to $x$, accumulating probability mass up to $x$.

Example in a business

$P(\text{session duration} > 5 \text{ min}) = 1 - F(5)$ where $F$ is the CDF of the fitted lognormal distribution. Used to set SLA thresholds or product performance benchmarks.

The formula, piece by piece

$F(x) = \int_{-\infty}^{x} f(t) \, dt$

• $F(x)$: cumulative probability up to $x$.
• $f(t)$: the PDF.
• Derivative: $F'(x) = f(x)$ (the PDF is the derivative of the CDF).

The formula in words

"The CDF at $x$ is the total area under the PDF curve from negative infinity to $x$."

Critical Concepts

• $F'(x) = f(x)$: PDF = derivative of CDF. CDF = integral of PDF.
• $P(a \leq X \leq b) = F(b) - F(a)$.
• Quantile function: $Q(p) = F^{-1}(p)$ — inverse of CDF.
• Python: scipy.stats.<dist>.cdf(x), inverse: ppf(p).
• PIT: if $X \sim F$, then $F(X) \sim \text{Uniform}(0,1)$ — used for residual diagnostics.

Step-by-step calculation (tiny example)

Step 1: Exponential distribution, rate λ=1 PDF: $f(x) = e^{-x}$ for $x \geq 0$.

Step 2: CDF integral $F(x) = \int_0^x e^{-t} dt = [-e^{-t}]_0^x = 1 - e^{-x}$

Step 3: Query $P(X \leq 2) = 1 - e^{-2} \approx 0.865$

Code examples

Beginner example (Python)

from scipy.stats import expon, norm
import numpy as np   # Exponential: P(X <= 2) where lambda=1 (scale=1/lambda=1)
p_under_2 = expon.cdf(2, scale=1)
print(f"P(Exp(1) ≤ 2) = {p_under_2:.4f}")   # ~0.865 # Normal: 95th percentile (inverse CDF)
p95 = norm.ppf(0.95, loc=0, scale=1)
print(f"95th percentile of N(0,1) = {p95:.4f}")   # ~1.645

Beginner example (R)

cat(sprintf("P(Exp(1) <= 2) = %.4f\n", pexp(2, rate=1)))
cat(sprintf("95th percentile N(0,1) = %.4f\n", qnorm(0.95)))

Professional example (Python)

import numpy as np
from scipy.stats import lognorm, kstest
from scipy.special import erfc

def fit_and_evaluate_cdf( data: np.ndarray, distribution_name: str = "lognorm" ) -> dict[str, float | bool]:
    """Fit a distribution and evaluate CDF for SLA analysis. Uses probability integral transform (PIT) check: if model is correct, F(data) ~ Uniform(0,1). Args: data: Positive continuous data. distribution_name: scipy.stats distribution name. Returns: Dict with fit params, SLA probabilities, and PIT KS test result. """
    import scipy.stats as ss
    dist = getattr(ss, distribution_name)
    params = dist.fit(data, floc=0)
    shape_args = params[:-2]
    loc, scale = params[-2], params[-1]
    fitted = dist(*shape_args, loc=loc, scale=scale)
    pit_values = fitted.cdf(data)
    ks_stat, ks_p = kstest(pit_values, "uniform")
    return { "p_under_p50": float(fitted.cdf(np.percentile(data, 50))), "p95_quantile": float(fitted.ppf(0.95)), "ks_pit_statistic": ks_stat, "ks_pit_pvalue": ks_p, "distribution_adequate": ks_p > 0.05, }
rng = np.random.default_rng(seed=42)
session_durations = rng.lognormal(mean=1.5, sigma=0.8, size=2_000)
result = fit_and_evaluate_cdf(session_durations)
print("Lognormal CDF fit:")
for key, val in result.items():
    print(f" {key}: {val}")

Source: domain knowledge

#probability-statistics#statistical-operations-and-random-variables#continuous-random-variables-crvs