Probability Theory and mathematical statistics: random variables, distribution law, distribution function, Density Function

1: For N results in one experiment, if a real-value single-value function is matched for each result, the corresponding value is called a random variable. The mathematical definition is S = {e}. E is a sample in the sample space, and X = x (e) is called a random variable, in fact, it is to map the results of an experiment to a numerical value. This value is called a random variable.

2: random variables include continuous random variables and continuous random variables. You can see how to differentiate them by name. The concept of different random variables is also different.

3: Distribution Law: a two-dimensional table that describes the probability of the event corresponding to the random variable and the random variable. For example, the random variable values include 0, 1, 2, the probability of events corresponding to each random variable (a random variable may correspond to multiple events) is 0.1, 0.5, and 0.4, respectively. Then the distribution law is

X	0	1	2
P	0.1	0.5	0.4

As you can see from the figure, only discrete random variables have a distribution law. Continuous Random Variables have too many values and cannot be drawn.

4: Distribution Function: defines a mathematical formula for exporting. F (x) = P {x <= x}, then f (x) is called the distribution function of random variable X (pay attention to uppercase X and lowercase X ), distributed functions are applicable to discrete and continuous random variables.

5: The distribution function can define something similar to the distribution law for continuous random variables. See the definition: If f (x) = integral symbol F (x) dx, X becomes a continuous random variable. f (x) is called a density function. In fact, when I understand this, we can look at the distribution law of discrete random variables.

6: Based on some of the above concepts, some common distributions can be derived.

6.1: the typical distribution of discrete random variables is 01 (there are only two types of random variables, and this is done once in the experiment), benuli (random variables are only 2 in the middle, however, the experiment is performed n times), Poisson distribution (the probability of each test is... add it here. The csdn formula is too troublesome)

6.2: the common distribution of continuous random variables is: even distribution (that is, the density function satisfies ...), exponential Distribution (the density function is satisfied), normal distribution (the density function is satisfied ...)

The specific formulas 6.1 and 6.2 are not provided. The purpose of listing these formulas is to compare how to remember these distributions. Discrete is based on the probability of each test and the number of experiments performed. Continuous Distribution relies on distribution functions to differentiate different distributions.