Understanding of the general and the sample

For this part of the understanding there is always some uncertainty, so in addition to the use of formulas, sets of models, almost can not be extended to understand.

Read the most basic definition, found very simple and easy to understand. Excerpt + understand below.

In the Probability theory section, we have already known the distribution model of a variable, with this model to study or to calculate the corresponding properties, digital features and so on.

But in mathematical statistics this part, the angle of the thing is not the same: the study of random variable distribution is not known, but we often need to use some means to infer the distribution of the random variable. This method is: Repeated observations of several independent repetitions, to obtain some observation values. Based on these observations, we can make a reasonable inference to the distribution of this variable.

First look at what we need to study. For example, to investigate the height of a school's students, the quantitative indicator of height is the object of study. The height of the school students is a random variable, it may be set to x, variable to the value range, where the range is the height of the whole school students value. X is the whole, also called the Total X. It is particularly emphasized that overall x is a random variable that expresses all values. The overall is divided into a limited whole and an infinite whole . Studying the height of a school student is naturally a limited whole. If the object is the life of a certain type of lamp that is being used nationwide, the number of observations can be considered as an infinite whole.

The overall distribution is unknown, it has some form that can be modeled, and secondly, the model contains an unknown parameter, which will be inferred by sampling.

The part of the individual being extracted is called a sample of the population.

To extract an individual from the whole is to observe the total X and record the results . Under the same conditions, the population was repeatedly observed N times. The n observation results were recorded as x1,x2,.., Xn x_1,x_2,.., x_n in the experimental sequence. They are independent of each other and are random variables that have the same distribution as X . x1,x2,.., Xn x_1,x_2,.., X_n called a simple random sample from the overall X.

In particular, it is necessary to understand that each of the values in the sample is considered a random variable, because the randomness of the sampling, so that each individual is a response to the overall, so and the overall x is a level, such as the total X value range, in each individual, the value range is the same.

For the sampled samples, we can get the observed value x1,x2,..., xn x_1,x_2,..., x_n, which is called the sample value.

The extraction process is also divided into a back-up and non-return sampling. It is inconvenient to put back the sampling, so it is approximate to replace the sample with no back-up. Of course, there is no difference between the infinite population, not approximate.

Strict definition : Set X is a random variable with distribution function f, if x1,x2,.., Xn x_1,x_2,.., x_n is a random variable with the same distribution function f, independent of each other, it is said x1,x2,.., Xn x_1,x_2,.., x_n is A simple random sample with a capacity of n from the distribution function f (or population f, or total x) , referred to as a sample. The observed value x1,x2,..., xn x_1,x_2,..., x_n is called a sample value and is also an n independent observation of X.

Then: f∗ (x1,x2,..., xn) =∏ni=1f (xi) f^* (x_1,x_2,..., x_n) = \PROD_{I=1}^NF (x_i)

If x has a probability density f f, then: