### Probability density function commonly understood

Probability distribution of continuous random variables

In the beginning, stupidly can not distinguish the specific meaning of these two concepts. The literal meaning of the feeling is not too much difference, in fact, he said the actual meaning of the difference is really not too much, just on the independent variable interval of different different names, and different ways of calculating.

First of all, the introduction of the concept of random variables, the variable can be subdivided into discrete random variables and continuous random variables.

Discrete variable: If you provide a unit length of 1 meter, so that you take a scale every 10mm, then the length of which the value of the value of the discrete variable.

Continuous variable: if you provide a unit length of 1 meter, so that you are free to choose, without limiting the interval, then you can take an infinite number of corresponding length values. The variable in this case we can understand as a continuous variable.

The probability distribution function and the probability density function are both probability functions. So what is a probability function?

Probability function, refers to the expression of probability in the form of a function.

Probability distribution functions and probability density functions are nothing more than functions used to describe the magnitude of the probability of an event occurring at a certain point or in a certain interval. Classifying them as probability distribution and probability density functions is essentially a categorical discussion of continuous and discrete variables, specific values, specific analysis. The probability distribution function and the probability density function of the full interval of the result must be both 1, that is, the event must occur in the full interval segment.

### What is the difference between a probability density function and a probability function

I think you are talking about a probability density function and a probability distribution function.

Assuming X is a variable, the probability distribution function F(x)=P{X<x}, cumulativedistributionfunction.

If X is discrete, then P(X=x) is defined as the probability mass function (probabilitymassfunction)

If X is continuous, then there exists a probabilitydensityfunction,which is the derivative of the probability distribution function

f(x)=F'(x).

### What is the relationship between the probability density function and probability?

The probability density function is for continuous random variables, suppose for a continuous random variable X, its distribution function is F(x) and the probability density is f(x).

The probability density can be calculated along the following lines:

F(x)=∫[-∞,x] by definition.

f(y)dy can be seen F'(x)=f(x), that is, the derivative of the distribution function is equal to the probability density function, so you only need to find the derivative on the basis of the original distribution function to get the probability density function.

Distribution function

It is an important function in probability statistics, and it is through it that mathematical analysis can be used to study random variables. The distribution function is the most important probabilistic feature of a random variable, and the distribution function provides a complete description of the statistical laws of a random variable and determines all other probabilistic features of the random variable.

### Difference between probability density and probability. Why Probability Density Can Be Greater Than 1

Probability density alone is meaningless because it must involve range.

Probability density * range = probability before probability fits into that concept of not being greater than 1.

The area of a normal distribution image is 1, representing the sum of the probabilities of all events.

And the probability density helps you to accurately calculate the probability of a certain interval, that’s what it’s there for, there’s not much point at all in discussing whether it’s greater than 1 or not, and it doesn’t represent the probability of a certain range.

### What is the probability density function?

Set: the probability distribution function is: F(x)

The probability density function is: f(x)

The relationship between the two is: f(x)=dF(x)/dx

That is, the density function f is the first-order derivative of the distribution function F. Or the distribution function is the integral of the density function.

The distribution function is defined because in many cases we do not want to know the probability of something being at a particular value, but at best we want to know the probability of it being in a certain range, and so the concept of the distribution function is introduced.

And the probability density, if continuous at x. It is the distribution function F(x) that is derived from x. Conversely, knowing the probability density function, the distribution function can be derived by integrating negative infinity to x.

Probability density:

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

Reference: Baidu Encyclopedia-Probability Density