Todennäköisyysdensiteettifunktio

todennäköisyysdensiteettifunktio, or probability density function (PDF), is a fundamental concept in probability theory and statistics, particularly for continuous random variables. It describes the relative likelihood for a continuous random variable to take on a given value. The value of the PDF at a particular point does not represent the probability of the variable being exactly that value, as the probability of any single point for a continuous variable is zero. Instead, the area under the PDF curve between two points represents the probability that the variable falls within that range.

Mathematically, for a continuous random variable X, its probability density function is denoted by f(x). The integral of f(x) from negative infinity to positive infinity must equal 1, representing the certainty that the variable will take on some value. Similarly, the integral of f(x) from a to b gives the probability P(a <= X <= b). Key properties include that the PDF is always non-negative, i.e., f(x) >= 0 for all x, and its integral over the entire domain is 1. Different probability distributions, such as the normal distribution or the uniform distribution, have distinct PDFs that characterize their behavior. Understanding the PDF is crucial for calculating probabilities, moments, and making statistical inferences about continuous data.