A probability density function (PDF) is a fundamental concept in statistics and probability theory, particularly for continuous random variables. It describes the likelihood of a random variable taking on a specific value. The PDF is derived from the cumulative distribution function (CDF), which represents the probability that a random variable is less than or equal to a certain value. The PDF itself does not provide probabilities directly; instead, it gives a probability density, which must be integrated over an interval to obtain a probability mass.

For example, the PDF of a normal distribution is expressed mathematically as:

[ \text{PDF}_{\text{normal}}(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} \left(\frac{x - \mu}{\sigma}\right)^2} ]

where ( \mu ) is the mean and ( \sigma ) is the standard deviation. Evaluating a PDF at a specific point does not yield a probability; rather, it indicates how dense the probability is around that point. To find the actual probability of a random variable falling within a certain range, one must integrate the PDF over that range ¹²³.