Densitetsfunktionen

Densitetsfunktionen, also known as probability density functions (PDFs), are fundamental concepts in probability theory and statistics. They describe the likelihood of a continuous random variable taking on a particular value. Unlike probability mass functions, which are used for discrete random variables, PDFs are used for continuous random variables.

A PDF, denoted as f(x), has the property that the probability of a random variable X falling within a certain range [a, b] is given by the integral of the PDF over that range. Mathematically, this is expressed as P(a ≤ X ≤ b) = ∫ from a to b f(x) dx. The total area under the PDF curve over the entire range of the random variable is equal to 1, reflecting the fact that the total probability must sum to 1.

The mean (expected value) of a continuous random variable X with PDF f(x) is calculated as the integral of x times the PDF, weighted by the PDF itself. Similarly, the variance is calculated as the integral of the squared deviation from the mean, also weighted by the PDF.

Common examples of PDFs include the normal distribution, which is characterized by its bell-shaped curve, and the exponential distribution, which is often used to model the time between events in a Poisson process. Each of these distributions has its own unique properties and applications in various fields, such as finance, engineering, and natural sciences.