Probability density and distribution functions

The formula

P(a ≤ X ≤ b) = ∫[a to b] f(x) dx The probability density function f(x) only produces a probability through area. Height alone means nothing.

f(x) ≥ 0 for all x, and ∫[−∞ to +∞] f(x) dx = 1 A density can never go negative, and the total area is exactly 1 — every outcome lands somewhere.

F(x) = P(X ≤ x) = ∫[−∞ to x] f(t) dt The cumulative distribution function: the fraction of outcomes at or below x. Runs from 0 to 1, never decreases ^[2].

f(x) = dF/dx The density is the slope of the CDF. Steep CDF means outcomes pile up there.

P(X > x) = 1 − F(x) Exceedance, also called the survival or reliability function. This is the number a spec limit or a failure budget actually depends on.

p(k) = P(X = k), Σ p(k) = 1 For discrete variables — failure counts, bit errors, parts per lot — the probability mass function assigns probability to each value directly. Sum, don't integrate ^[1].

f(x) = (1/(σ·√(2π))) · exp(−(x − μ)²/(2σ²)) The normal (Gaussian) density, the default model for measurement scatter: mean μ sets the center, standard deviation σ sets the spread ^[3].

Where you meet it

Setting acceptance limits on a production ATP. Your gain measurement scatters around nominal with some σ. Put the limit at μ − 3σ and the normal CDF says 0.135% of good units fall below it — about 1 in 741. Put it at μ − 2σ and you scrap 2.3% of hardware that was probably fine. The limit is a point on a CDF whether you drew the curve or not.
Bearing and component life on a test stand. Reliability engineers quote L10 life: the time at which the Weibull CDF F(t) = 1 − exp(−(t/η)^β) reaches 0.10 ^[4]. For a bearing with shape parameter β = 1.5, L10 sits at about 22% of the characteristic life η. Ten percent of the population is gone by then — L10 is a percentile, not a promise.
The RF bench. Thermal noise voltage is Gaussian. The PDF tells you the instantaneous amplitude statistics, which is why crest factor matters: a Gaussian signal spends most of its time near zero but will eventually exceed any threshold you name. How often is an exceedance calculation, 1 − F(x), straight off the normal CDF.
Random vibration review. Shaker controllers clip drive signals at 3σ. Someone at the table will claim that guts the signal. The normal statistics say otherwise: instantaneous acceleration exceeds ±3σ only 0.27% of the time, the tail beyond ±3σ carries about 2.9% of the variance, and hard clipping there removes only about 0.5% of the total energy. And when the question shifts to a spec limit on one side, keep one-sided versus two-sided straight — 0.135% out on the guarded side, 0.27% out in total.

How it works

The central fact: density is not probability. For a continuous variable, P(X = exactly x) = 0 — probability lives only in intervals, as area under f. The density carries units of 1-per-x-unit (per MPa, per volt, per hour), and it can legitimately exceed 1 anywhere the distribution is narrow enough. A normal density with σ = 0.1 peaks at about 4.0. Nothing is wrong; the area is still 1. Reading PDF height as "probability of that value" is the single most common misuse of a distribution curve.

The CDF is the workhorse. Every practical question — what fraction fails low, what fraction exceeds the limit, what value bounds the worst 1% — is a CDF or inverse-CDF lookup. The PDF is for judging shape: is the scatter symmetric, skewed, or bimodal (bimodal usually means two populations got mixed, which is a process finding, not a statistics finding). If you catch yourself integrating a PDF numerically to answer a fraction-out-of-spec question, stop and use the CDF that software already provides.

Percentiles come from running the CDF backwards. Solve F(x_p) = p for x_p and you have the value that fraction p of the population sits below. That inversion is all an L10 life, a 95th-percentile latency, or a "99% of gusts below this load" requirement is.

The gotcha that bites hardest: models get fit in the middle and used in the tails. Twenty or thirty data points will nail down μ and σ nicely, and a normal curve will lie beautifully over the histogram between ±2σ. That fit tells you almost nothing about behavior at ±4σ, where your one-in-ten-thousand spec actually lives. Real processes have truncation, contamination, and physical limits the Gaussian doesn't know about. If a tail probability drives a decision, either bring enough data to see the tail or use a distribution with a physical argument behind it — Weibull for weakest-link failures, exponential for constant-hazard electronics, Rayleigh for the envelope of Gaussian noise.

Two smaller traps. Histograms estimate a PDF only after you divide counts by both sample size and bin width; overlay a normal curve on raw counts and the comparison is meaningless. And keep one-sided versus two-sided straight: 68.27% of a normal population sits within ±1σ, 95.45% within ±2σ, 99.73% within ±3σ — so "3σ" means 0.27% out in total but only 0.135% out on the side your spec limit guards.

History

The bell curve showed up before anyone called it a distribution. In November 1733 Abraham de Moivre, a French refugee doing mathematics for a living in London coffeehouses, circulated a short Latin pamphlet showing that the binomial distribution — coin-flip arithmetic that became hopeless for large n — could be approximated by a smooth exponential-of-a-square curve. He folded the result into later editions of his textbook The Doctrine of Chances, and it stands as the first appearance of the normal probability integral ^[5]^[6].

The curve became an engineering tool through astronomy. Gauss had used his least-squares method to recover the orbit of Ceres in 1801, and published the method in 1809 in his treatise on celestial mechanics, Theoria motus corporum coelestium; Legendre had published least squares independently in 1805 ^[7]^[8]. Treating observation error as a random variable with a density — and choosing the density that made least squares the right answer — is why the curve carries Gauss's name.

The modern definitions arrived in 1933, when Andrey Kolmogorov's monograph Grundbegriffe der Wahrscheinlichkeitsrechnung rebuilt probability from axioms the way Euclid built geometry, giving the distribution function a rigorous footing instead of a working intuition ^[9]^[10].

The version reliability engineers use daily came from one of their own. Waloddi Weibull, a Swedish engineer and materials scientist, published his statistical theory of material strength in 1939 and in 1951 presented "A statistical distribution function of wide applicability" to ASME, backing the claim with seven case studies ^[11]^[12]. His CDF, chosen for flexibility rather than theoretical purity, became the standard model for fatigue and bearing life — which is why the 10% point on a CDF is stamped on bearing catalogs as L10.

Related tools

/tools/bearing-life-l10 — L10 is the 10th percentile of a Weibull CDF
/tools/crest-factor — peak-to-RMS behavior is tail behavior of the amplitude PDF
/tools/thermal-noise-floor — Gaussian noise statistics in dBm terms
/tools/rms-peak — RMS is the square root of the second moment of the density
/tools/snr-enob — quantization noise modeled as a uniform density
/tools/half-life-decay — the exponential survival function, R(t) = exp(−λt), in decay form

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →