Normal PDF f(x; μ,σ)

Normal PDF: density of the bell curve

The probability density function of the Gaussian is f(x) = (1 / (σ√(2π)))·e^(−(x − μ)² / (2σ²)). Its peak sits at the mean μ, the curve bends at μ ± σ, and the whole thing integrates to 1 over ℝ. What you read off the PDF is density, not a probability. To get an actual probability you measure the area under the curve. Standardise it and you get φ(z) = (1/√(2π))·e^(−z²/2), whose peak is about 0.3989 at z = 0. As a quick check, with μ = 0, σ = 1 and x = 1 you get f(1) ≈ 0.2420. The familiar 68-95-99.7 rule still applies: roughly 68% of the mass falls inside μ ± σ, 95% inside μ ± 2σ, and 99.7% inside μ ± 3σ.

Applications

It shows up in biometric data such as height and IQ, in measurement error across physics and engineering, and in finance, where Black-Scholes leans on log-normal returns. Statistical process control uses it too, with Six Sigma aiming at 3.4 defects per million. Regression depends on it when OLS treats residuals as normal, as do the likelihood functions behind maximum-likelihood estimation.

FAQ

Can f(x) be greater than 1? It can. Remember that density isn't probability. Once σ drops below 1/√(2π) ≈ 0.399, the peak climbs past 1. The thing that always equals 1 is the total integral.

Where are the inflection points? They sit at μ − σ and μ + σ, the spots where curvature flips sign. That makes them handy for reading σ straight off a plotted bell curve.

PDF vs CDF? The PDF tells you the density at a single point. The CDF tells you the probability that has piled up so far, P(X ≤ x). One is the integral of the other.

When does the normal fail? It struggles with skewed data like income or time-to-failure, and with heavy tails of the sort you see in financial crashes. In those cases reach for a log-normal, Student's t, or generalized extreme value distribution instead.