### Difference between the probability density function and the distribution function

1, the probability density function is a description of the output value of this random variable, in the vicinity of a certain value point of the possibility of function. The probability that the value of the random variable falls within a certain region for the probability density function in this region of the integral, when the probability density function exists, the cumulative distribution function is the integral of the probability density function, the probability density function is generally marked in lowercase;

2, the distribution function is the probability of statistics in the important function, through the function can be used in mathematical analysis of the method of studying the random variable, the distribution function is the random variable, the distribution function is the random variable, the distribution function is the probability that the value of the random variable falls near a certain point. Random variable, the distribution function is the most important probability characteristics of random variables, the distribution function can be a complete description of the statistical laws of random variables, and to determine all other probability characteristics of random variables.

### May I ask what is the difference between probability density, distribution function, and distribution law?

To make it simple, the details are shown in the picture

### Difference between Probability Density Function and Distribution Function

Mathematically, the distribution function, F(x)=P(X<x), represents the probability that the value of a random variable, X, is less than x. The meaning is easy to understand.

The probability density f(x) is the first-order derivative of F(x) with respect to x at x, the rate of change. If you take a very small neighborhood Δx around a certain x, then the probability that the random variable X falls within (x,x+Δx) is approximately f(x)Δx, i.e., P(x<X<x+Δx)≈f(x)Δx.

In other words, the probability density f(x) is the probability that X falls within a “unit width” at x “in x. The term “density” can be understood in this way.

### Difference between probability density function and distribution function

Difference between probability density function and distribution function?

Answer:

The probability density function is the derivative of the distribution function.

The distribution function is the integral of the probability density function with a final value equal to 1.

### Difference and Connection between Distribution Function and Density Function

The difference and connection between DistributionFunction and DensityFunction are as follows:

DistributionFunction and DensityFunction are two concepts that are commonly used in probability theory and statistics to describe the distribution of a random variable. Although they are somewhat similar, there are some differences and connections in their definitions, properties and applications.

1, definition: distribution function: for a random variable X, its distribution function F (x) is defined as F (x) = P (X ≤ x), indicating that the random variable X is less than or equal to x probability. Density function: for a continuous random variable X, its density function f(x) is defined as the probability of ∫f(x)dx on any interval [a,b], i.e., P(a≤X≤b)=∫f(x)dx.

2. Properties: Distribution function: F(x) is a monotonically non-decreasing function. When x tends to negative infinity, F(x) tends to 0; when x tends to positive infinity, F(x) tends to 1. F(x) is right-continuous, i.e., lim┬(h→0⁺)F(x+h)=F(x). Density function: f(x) is nonnegative and unique for any x on the real number axis. The integral of f(x) over the entire real number axis is equal to 1, i.e., ∫f(x)dx=1. For any interval [a,b], the probability is equal to the area under the curve of the density function under that interval.

3. Connection: for continuous random variables, the distribution function F(x) and the density function f(x) are closely related. The distribution function is defined through the density function. For a continuous-type random variable X, the derivative f(x) = dF(x)/dx is its density function. Conversely, the integral F(x)=∫f(x)dx is its distribution function. The distribution function gets its probability by integrating the density function, i.e., P(a≤X≤b)=∫[a,b]f(x)dx=F(b)-F(a). The density function can be obtained by derivation of the distribution function, i.e., f(x)=dF(x)/dx.

So the distribution function and the density function are two probabilistic representations describing the distribution of a random variable. The distribution function is defined as the probability that a random variable is less than or equal to a certain value, while the density function is defined as the probability density over an interval. The two are related to each other through the relationship between derivatives and integrals; the density function is the derivative of the distribution function, while the distribution function is the integral of the density function. By transforming the distribution function and the density function into each other, we can calculate the probability and statistical properties of random variables.

The advantages of learning function

1, abstract thinking ability: learning function can develop abstract thinking ability, because function is an abstract mathematical concept. By learning functions, we can abstract concrete problems into the relationship between symbols and variables, so as to better understand and solve problems.

2. Solving practical problems: functions have a wide range of applications in real life. Understanding and learning about functions can help us solve a variety of practical problems, such as modeling, prediction, optimization and so on. The concepts and methods of functions can be applied in various fields, including science, engineering, economics, computer science, etc.

3, describe the relationship: functions can be used to describe the relationship between variables. Through functions, we can analyze and explain the dependencies between variables and understand their characteristics and trends. Functions can help us understand and predict the pattern of change in data, charts, images, etc.

4, analyzing and reasoning ability: learning functions can exercise our analyzing and reasoning ability. By analyzing the properties, characteristics and images of functions, we can deduce and prove mathematical conclusions and develop logical thinking and reasoning ability. This ability is important for solving complex problems and promoting academic research.

### Distribution function and probability density function of the difference

distribution function, that is to say, is a function of probability, in simple terms is f (x), x every value, f corresponds to the result of a probability

density function, that is to say it is the density of the probability of the response to the rate of change of the probability of it is the distribution function of the derivative, you can also be understood as it corresponds to the negative infinite to x The integral is f(x)