Is the probability distribution normal? What is a Normal Distribution Probability Table?

A normal distribution is a type of continuous probability distribution.

Probability distribution is one of the basic concepts of probability theory. It is used to express the probability law of the value of a random variable. There are different forms of probability distributions to describe different types of random variables.

Random variables can be divided into discrete and continuous.

1.Distribution Columns of Discrete Random Variables Random variables that take only a finite number of real values or can be listed are called discrete random variables. For example, if there are 10 defective products out of 100 products, and 5 of them are randomly selected, the number of defective products x is a discrete random variable that only takes 0, 1, 2, 3, 4, and 5.

To describe the probability distribution of a discrete random variable, use a distribution column, which gives all the values of the discrete random variable, and the probability of taking each value. For example, in the above example, the distribution of the number of defective products x is listed as: where represents the number of combinations of m from n different things:

2.Density function of a continuous random variable If there is a non-negative real function p(x) such that the distribution function f(x) of the random variable x can be expressed as the integral of p(x) to x, then x is called a continuous random variable, and p(x) is called the density function of x. The probability of taking any real value of a continuous random variable is equal to 0.

The common distributions of continuous random variables are: uniform distribution, normal distribution, Cauchy distribution, lognormal distribution, exponential distribution, gamma ( ) distribution, beta ( ) distribution, x2 distribution, student distribution, f distribution, and so on. The concept of distribution function is extended to the case of random vectors, and the concepts of joint distribution function, edge distribution function, joint distribution column, edge distribution column, joint density function and edge density function are obtained.

A normal distribution is just a type of probability distribution, which is a one-dimensional random variable distribution and a multi-dimensional random variable probability distribution.

It can be divided into the probability distribution of discrete random variables and the probability distribution of continuous random variables.

The normal distribution is only a kind of continuous random variable distribution, and the continuous random variable distribution also includes uniform distribution, exponential distribution, etc.

You can refer to the textbook "Probability Theory and Mathematical Statistics" in science and engineering universities.

Normal distribution is only one of them, so not necessarily!

Normal distribution, also known as ".Normal distribution", akaGaussian distribution(Gaussian Distribution), first sought by Abraham de MoivreBinomial distributionasymptotic formula. Gauss derived it from another angle when studying measurement errors.

Laplace. And Gauss studied its properties. It is a probability distribution that is very important in mathematics, physics, and engineering, and has a significant impact on many aspects of statistics.

The normal curve is bell-shaped, low at both ends and high in the middle, symmetrical to the left and right, because its curve is bell-shaped, so people often call it a bell-shaped curve.

If the random variable x obeys a mathematical expectation.

is a normal distribution with a variance of 2 and is denoted as n( ,2). Its probability density function is the expected value of the normal distribution that determines its position, its standard deviation.

determines the magnitude of the distribution. The normal distribution at =0, 1 is the standard normal distribution.

Theorem:

Since the image of a normal population is not necessarily symmetrical with respect to the y-axis, for any normal population, the probability that its value is less than x. As long as you can use it to find the probability that the normal population is in a particular interval.

In order to facilitate the description and application of the bent nucleus punch, normal variables are often used as data transformations. Converts the general normal distribution to the standard normal distribution.

If. Obeying the standard normal distribution, the probability value of the original normal distribution can be directly calculated by checking the standard normal distribution table. Therefore, this transformation is called a normalized transformation.

Standard Normal Distribution Table: The standard normal distribution table lists the proportional buried area from - to x (current value) under the standard normal curve. ）

Sigma principle: the probability of a numerical distribution in ( - is;

2sigma principle: the probability of a numerical distribution in (-2, 2) is;

3sigma principle: the probability of the numerical distribution in (-3 , 3 ) is;

where in the normal distribution represents the standard deviation and represents the mean x= which is the axis of symmetry of the image.

Due to the basic idea of "small probability event" and hypothesis testing "small probability event" usually refers to an event with a probability of occurring less than 5%, and it is considered almost impossible for the event to occur in a single experiment.

It can be seen that the probability of x falling outside (-3, 3) is less than 3 per thousand, and in practical problems, it is often thought that the corresponding event will not happen, and the interval (-3, 3) can be regarded as the actual possible value range of the random variable x, which is called the "3" principle of normal distribution.

The normal distribution is the distribution of a continuous random variable with two parameters and 2, the first parameter is the mean of the random variable that follows the normal distribution, and the second parameter 2 is the variance of this random variable, so the normal distribution is denoted as n( ,2 ). The probability law of a normally distributed random variable is that the probability of taking a nearby value is large, and the probability of taking a value farther away is smaller. The smaller it is, the more concentrated it is in the vicinity, and the larger it is, the more dispersed it is. The characteristics of the normally distributed density function are:

With regard to symmetry, the maximum value is reached at , the value is 0 at positive (negative) infinity, and there is an inflection point at . It is shaped with a high middle and low sides, and the image is a bell-shaped curve above the x-axis. When 0, 2 1 is called the standard normal distribution and is denoted as n(0,1).

When a dimensional random vector has a similar probability law, the random vector is said to follow a multidimensional normal distribution. For example, the marginal distribution of a multivariate normal distribution is still normally distributed, the random vectors obtained by any linear transformation are still multidimensional normal distributions, and especially its linear combination is a univariate normal distribution.

Normal distribution probability calculation formula: f(x)= x- ) Normal distribution is also called "normal distribution", also known as Gaussian distribution, the normal curve is bell-shaped, low at both ends, high in the middle, symmetrical left and right because its curve is bell-shaped, so people often call it a bell-shaped curve.

Main features: Estimating the frequency distribution A variable that obeys a normal distribution can estimate the proportion of frequencies in any value range according to the formula as long as the mean and standard deviation are known.

The normal distribution method is suitable for indicators that obey a normal (or near-normal) distribution, as well as those that can be converted to a normal distribution.

The percentile method is often used as an indicator of skewed distributions. The unilateral and bilateral cut-offs of the two methods in Table 3-1 should be mastered proficiently.

Quality control: In order to control the measurement (or experiment) error in the experiment, it is often used as the upper and lower warning value, and as the upper and lower control value. This is done on the basis that the measurement (or experimental) error follows a normal distribution under normal circumstances.

Normal distribution is the theoretical basis of many statistical methods. A variety of statistical methods, such as testing, analysis of variance, correlation vibrillation, and regression analysis, all require that the indicators of the analysis obey a normal distribution. Although many statistical methods do not require the analysis indicators to obey the normal distribution, the corresponding statistics are approximately normal distribution in large samples, so these statistical inference methods are also based on normal distribution in large samples.

How to calculate normal distribution probability in excel.

The syntax of the normal distribution function is normdist(x,mean,standard dev,cumulative), cumulative is a logical value, if 0 is the density function, if it is 1, it is the cumulative distribution function. If you plot a normal distribution, it is 0.

For example, a normal distribution with a mean value of 10% and a standard value of 20%.

First, type in a variable in A1, assume 50, select column a, point fill sequence, select column, equal difference series, step value 10, stop value 70.

Then type normdist(a1,10,20,0) in b1, and the return value is, b1 is selected;

When the mouse turns into a black cross in the lower right corner, scroll down to B13, select the A1B13 area, click the Chart Wizard on the toolbar Scatter Chart, select the second figure in the first row, click Next, Default Settings, Next, write the title by yourself, remove the tick in the grid line, remove the tick in the legend, click Next, and Done.

The diagram is preliminarily completed. The following is to fine-tune the mouse to right-click on the coordinate axis of the graph, select the coordinate axis format, fill in the scale you want to minimize the value, the maximum value, the main scale unit (the numerical interval on the x-axis), the y-axis intersection (when y is 0, how much x) and so on. Once determined, the normal distribution map is complete.

How to calculate the probability density function of a normal distribution.

Calculate the mean and standard deviation and substitute the expression of the normally distributed density function

f(x) =exp/[√2π)σ

Given the value of x, the f-value is calculated.

Probability calculation of normal distribution, x n(50,100), find p(x<=40).

A general calculation method for finding a normal distribution.

Generally speaking. If the independent random variable x i n(a i,b i 2) i = 1,2 ,,.n

Then x 1+...x n obeys the normal distribution n(a 1+.a_n , b_1^2+..b_n^2)

This fact can be obtained from the probability feature function.

If you haven't learned it, you can get it by induction.

This is the case where the sum of two normal distributions is calculated and then generalized to n.

Let x=. Five screw samples were taken from x, and the mean value of the samples was x'=(。

And brother J, fierce swim x n ( , sample mean x'~n(μ,n)。Point estimates'=x'=。

Normal distribution is the theoretical basis of many statistical methods: for example, the t-distribution, f-distribution, and x2 distribution are all derived from the normal distribution, and the u-test is also based on the normal distribution. In addition, the limits of the t-distribution, binomial distribution and Poisson distribution are normally distributed, and under certain conditions, they can be treated according to the principle of normal distribution.

There is also the central limit theorem, in the objective reality there are many variables of the blind machine, they are formed by the comprehensive influence of a large number of independent random factors, and each of these individual factors plays a small role in the overall influence, and the disadvantages of this random variable tend to obey the normal distribution approximately. This phenomenon is the objective background of the central limit theorem.