Tihedusfunktsioonide

Tihedusfunktsioonid, also known as probability density functions, are fundamental tools in probability theory and statistics for describing the distribution of continuous random variables. A tihedusfunktsioon assigns a non‑negative value to each point in the variable’s domain such that the total integral over the entire domain equals one, reflecting the certainty that the variable takes some value within that range. For a random variable X, the probability that X lies in an interval [a, b] is obtained by integrating its density function f over that interval: P(a ≤ X ≤ b) = ∫_a^b f(x) dx. This integral property ensures that tihedusfunktsioonid encode all information necessary to compute probabilities and expectations of functions of the random variable.

The concept contrasts with discrete probability mass functions, which assign probabilities to distinct outcomes and sum to one. While mass functions use summation to accumulate probability mass over individual points, density functions require integration because continuous outcomes have zero probability at any single point. Common examples of density functions include the uniform density on [0,1], where f(x)=1 for 0≤x≤1; the normal density, where f(x) = (1/√(2πσ²)) exp(-(x-μ)²/(2σ²)); and the exponential density with f(x) = λ exp(-λx) for x≥0. In many applications, such as signal processing, physics, and economics, tihedusfunktsioonid provide a compact representation of underlying random behavior and facilitate the derivation of moments, cumulative distribution functions, and other statistical descriptors.