The difference and connection of PDF, PMF and CDF in probability theory

In probability theory, PDF, PMF and CDF often appear, so what is the difference and connection between these three?

1. Concept explanation

PDF: Probability density function. In mathematics, the probability density function of a continuous random variable (it can be referred to as a density function if it is not confused) is a description of the output value of this random variable. A function of the probability near the value point.
PMF: probability mass function (probability mass function), in probability theory, the probability mass function is the probability of a discrete random variable at each specific value.
CDF: Cumulative distribution function (cumulative distribution function), also called distribution function, is the integral of probability density function , which can completely describe the probability distribution of a real random variable X.

2. Mathematical representation

2.1 PDF

If $X$ is a continuous random variable, the probability density function is defined as $f_X(x)$ , use the integral of PDF in a certain interval to describe the probability that a random variable falls in this interval, namely
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2.2 PMF

If $X$ discrete random variable, the probability mass function is defined as $f_X(x)$ , PMF is actually the distribution law of discrete random variables learned in high school, that is,
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for tossing a uniform coin, if heads make $X = 1$ , if the opposite is set $X = 0$ , then its PMF is

2.3 CDF

No matter what type of random variable (continuous/discrete/other), its cumulative distribution function can be defined, sometimes referred to as the distribution function for short.

For continuous random variables, there are obviously:

then CDF is the integral of PDF, and PDF is the derivative of CDF.
For discrete random variables, the CDF is a piecewise function. For example, the coin toss random variable in the example, its CDF is:

3. Concept analysis

Based on the above, we can get the following conclusions:

PDF is unique to continuous variables, PMF is unique to discrete random variables;
The value of PDF itself is not a probability, it is a trend (density). Only after integrating the value of a continuous random variable can it be a probability, that is to say, it is meaningless to determine the probability of a continuous value at a certain point;
The value of PMF itself represents the probability of that value.

4. The meaning of the distribution function

We analyze the meaning of the distribution function from two points:

4.1 Why do we need a distribution function?

For discrete random variables, you can directly use the law of distribution to describe its statistical regularity; for continuous random variables (non-discrete random variables), we cannot enumerate all possible values of random variables, so its Probability distribution cannot be described by the law of distribution like discrete random variables. So the PDF is introduced, and the integral is used to find the probability of a random variable falling into a certain interval .

The law of distribution (PMF) cannot describe continuous random variables, and the density function (PDF) cannot describe discrete random variables. Therefore, it is necessary to find a unified way to describe the statistical law of random variables, which has a distribution function .

In addition, in real life, sometimes people are interested in the probability that a random variable falls within a certain range. For example, if the number of dice is less than 3 points to win, then consider the probability that the random variable falls within a certain range. Becomes realistic, so it is necessary to introduce a distribution function.

4.2 The significance of the distribution function

Distribution function The function value of $F$ $($ $x$ $)$ at point x xx represents $X$ falls in the interval $(−\infty,x)$ , so the distribution function is defined asA common function of $R$ , so we can convert probability problems into function problems, so that we can use common function knowledge to study probability problems, which increases the scope of probability research.

5. References

PDF, PMF, CDF in probability
http://www.dataguru.cn/thread-150756-1-1.html
https://www.zhihu.com/question/23022012
https://www.zhihu.com/ question/36853661
https://www.zhihu.com/question/21911186
http://wenku.baidu.com/view/823a0bb9f111f18582d05a14.html

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