Una función de distribución, a menudo llamada función de distribución acumulada (CDF), is a fundamental concept in probability and statistics. It provides a complete description of the probability distribution of a random variable by detailing the likelihood that the variable will take on a value less than or equal to a specific point. In simpler terms, it allows us to understand how probabilities accumulate over a range of values.
Matemáticamente, para una variable aleatoria X, la función de distribución F(x) se define como:
F(x) = P(X ≤ x)
Esta ecuación indica que F(x) da la probabilidad de que la variable aleatoria X sea menor o igual al valor x. La función tiene varias propiedades importantes:
- No decreciente: A medida que x aumenta, F(x) no disminuye.
- Límites: F(x) approaches 0 as x approaches negative infinity and approaches 1 as x approaches positive infinity.
- Rango: Los valores de F(x) van de 0 a 1.
Distribution functions can be used in various applications, such as determining probabilities, making predictions, and performing statistical analyses. They are foundational in fields like aprendizaje automático, where understanding the distribution of data points is crucial for developing models. Additionally, different types of distribution functions exist, such as normal, binomial, and Poisson distributions, each describing different types of data behavior.