# How is the edge probability density function calculated

### How is the marginal probability density calculated?

Marginal probability densities are obtained by integrating the joint probability density function according to the range of the variable, obtaining the marginal probability density of the Y integral, and obtaining the marginal probability density of the X integral as follows:

The probability of a continuous random variable taking values at any point is 0. As a corollary, the probability of a continuous random variable taking values on an interval is independent of whether the interval is an open or closed interval is irrelevant. Note that the probability P{x=a} = 0, but {X=a} is a possible event.

Extension

The probability density function of a continuous random variable has the following properties:

If the probability density function fX(x) is continuous at a point x, then the cumulative distribution function is derivable and its derivative: dFx(x)/dx=fx(x).

Since the value of the random variable X depends only on the integral of the probability density function, the value of the probability density function at individual points does not affect how the random variable behaves.

More precisely, a function can also be a probability density function of X if there are only a finite, countably infinite number of points at which the function and the probability density function of X take different values, or if it measures 0 (is a zero-measure set) with respect to the entire axis of real numbers.

### Marginal probability densities:? How is it calculated at?

The marginal probability density function is a function that describes the probability distribution of a random variable X, and is used to represent the probability density at a particular value x. There are several ways to calculate the marginal probability density function, the most common of which is to solve it using the probability definition of probability theory, i.e.: f(x)=P(X=x), where P(X=x) denotes the probability that the random variable X takes on the value x.

### What is the edge density function?

The edge density function fx is equal to the result obtained by integrating f(x,y) over y.

And the conditional probability density is calculated on the basis of the marginal density function.

Implications

Then X is a continuous random variable, and f(x) is said to be the probability density function of X, or simply the probability density.

Simply speaking of the probability density has no practical significance; it must be predicated on a definite bounded interval. You can think of the probability density as a vertical coordinate, the interval as a horizontal coordinate, the probability density of the interval of the integral is the area, and this area is the probability of the event in the interval, all the area of the sum of 1. So analyzing the probability density of a point alone does not have any significance, it must have the interval as a reference and comparison.