- [[integral]], [[calculus - integrals applications]], [[relationship between probability density and cumulative distribution functions]], [[cumulative distribution function]], [[percent point function]] # Idea A function that describes probabilities for a set of exclusive **continuous** events ([[probability mass]] for discrete events). In the context of probability distributions, specifically for continuous random variables, the total area under the probability density function (PDF) should sum (or integrate) to 1 . This property ensures that the function indeed represents a valid probability distribution. Mathematically, for a probability density function $f(x)$, this condition is expressed as: $ \int_{-\infty}^{\infty} f(x) d x=1 $ This integral extends over all possible values of $x$, and the area it encloses corresponds to the total probability, which must be 1 to reflect the certainty that some outcome within the variable's range will occur. ![[Pasted image 20220102205829.png|800]] Probability density is the **rate of change in cumulative probability**. So where cumulative probability is increasing rapidly, density can easily exceed 1 (but if we calculate the area under the density function, it will never exceed 1). If $f(x)$ is continuous at $x$, then the [[probability density|probability density function]] is the [[derivative]] of the [[cumulative distribution function]]. ```r dnorm(0, 0, 0.1) # p(x = 0 | mu = 0, sd = 0.1) # 3.989423 x value with fastest rate of change for ECDF ``` # References - https://courses.kristakingmath.com/library/applications-of-integrals-66dccd29/110497/path/step/58109873/ - [udemy cohen - mass vs density](https://www.udemy.com/course/statsml_x/learn/lecture/20014624#overview)