The difference between distribution function and probability density function of random variables

Catalog

1、 Distribution function

2、 Probability density function

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1、 Distribution function

Distribution function It shows the probability that the value of the random variable falls on a certain interval , Is a non decreasing function .

set up X It's a random variable ,x Is any real number , function

Become X Distribution function of . The distribution function is an ordinary function , Its definition field is the whole real number axis . Geometrically, he represents random variables X Fall in real numbers x And x The probability on the left , Therefore, the landing point is x1 And x2 The probability between can be determined by F(x2)-F(x1) obtain . His The value range is [0,1], And it is a non decreasing function . It is easy to know that for discrete random variables ,F(x) Stepped .

2、 Probability density function

Probability density function f(x) Represents a random variable X At point x The density of probability distribution , reflect A random variable X take x The probability of nearby values . And Probability distribution function The relationship is as follows ：

namely F(x) And f(x) It is the relationship between integral and derivative . And satisfy ：

For example, the probability density function of normal distribution is ：

When , Time is the standard normal distribution ：