Glossary term
Probability Density Function
A function describing the relative likelihood of a continuous random variable taking values.
Definition
modelA function describing the relative likelihood of a continuous random variable taking values.
A probability density function, or PDF, describes the distribution of a continuous random variable. Probabilities are obtained by integrating the density over an interval, not by reading the density value at a single point. PDFs are central to uncertainty analysis, reliability, signal processing, measurement error, and simulation.
A probability density function describes how probability is distributed over a continuous variable. For a random variable X with density f_X(x), probability over an interval is:
The total area under the density must be one:
The density value itself is not a probability. It can be greater than one when the variable’s unit interval is small; only integrated area over an interval gives probability.
Engineering use
PDFs represent tolerances, measurement noise, material strength, load uncertainty, lifetime distributions, sensor errors, manufacturing variation, and environmental exposure. Monte Carlo simulation samples from PDFs to propagate uncertainty through a model. Reliability analysis uses the overlap between load and resistance distributions, or lifetime densities such as Weibull distributions, to estimate failure probability.
The units of a PDF are the inverse of the variable’s units. If X is measured in millimetres, f_X(x) has units of inverse millimetres. This is why changing units changes the numerical height of the density curve but not the probability area.
Model choice
Choosing a distribution requires evidence. Normal distributions are convenient but can assign impossible negative values to quantities such as strength or time to failure. Lognormal, Weibull, uniform, triangular, empirical, or mixture distributions may be more appropriate depending on the physics and data quality.
Common mistakes
A common mistake is to read the height of a PDF as the probability of that exact value. For a continuous variable, the probability at one exact point is zero. Another mistake is fitting a distribution to sparse data and then trusting tail probabilities far outside the observed range. A good uncertainty review states data source, units, distribution family, parameter estimation method, truncation limits, correlations, and sensitivity of conclusions to the chosen PDF.