Probability theory and mathematical Statistics-ch6-sample and sample distribution

In probability theory, the stochastic variables are assumed to be known, and the properties and digital characteristics of the study are studied.

In mathematical statistics, the distributions of the random variables studied are unknown or not fully known, and many observations are obtained by repeating independent experiments to infer the various possible distributions of random variables.

1. Random samples

Overall: All possible observations of the trial. = Sample Space

Individual: Each possible observation value. = Sample Point

Capacity: The number of individuals included in the population.

Limited overall

Unlimited Total

A general correspondence to a random variable x, the study of the whole is the study of random variable X. Therefore, the total and corresponding random variables are not distinguished, collectively referred to as the total X.

Sample: In mathematical statistics, people are extracted from the population by a subset of individuals, based on the data obtained to deduce the overall distribution, the extracted part of the individual is called a sample of the population.

To observe the whole, we will get a random variable X1, for the overall n repeated, independent observation, you will get n random variable x1,x2,...,xn, which n random variable x1,x2,...,xn is the total random variable x observation results. The X1,x2,...,xn is independent and has the same distribution as x, known as a simple random sample from the overall x . n is called the capacity of the sample. A set of real x1,x2,..., xn is a random variable x1,x2,...,xn, called a sample value, also known as n Independent observations of X.

2. Sample Distribution

Samples are the basis of statistical inference, but often do not directly use the sample itself, but rather a function constructed by the sample.

Statistics: Set X1,X2,...,XN is a sample from the overall x, G (X1,X2,...,XN) is its function, and G does not contain any unknown parameters, it is said that G (X1,X2,...,XN) is a statistic. Statistics are also a random variable.

G (x1,x2,..., xn) is the observed value of the statistic.

Common statistics:

Experience distribution function:

The empirical distribution function (empirical distribution functions) is a distribution function based on a sample. If set, is the population of the sample values, which are arranged in order of size, the distribution function is called

The empirical distribution function is the statistic corresponding to the overall distribution function.

The overall distribution function is f (x), the empirical distribution function of the statistic is FN (x), with FN (x) to infer F (x), and when n is large enough, Fn (x) converges to f (x) with a probability of 1.

The distribution of statistics is called the sampling distribution.

The following is a description of the distribution of several commonly used statistics from the normal population:

Chi-Square Distribution:

Set X1,x2,...,xn from the population of N (0,1), it is said that the statistic obeys the chi-square distribution of degrees of freedom N. Note as χ2~χ2 (n)

Degrees of freedom refers to the number of independent variables contained in the right side of the equation.

Chi-square distribution of the upper ª sub-site:

T distribution

Set X~n (0,1), y~x2 (N), and X, y independent, the random variable is said to obey the t distribution of degrees of freedom. Note as T~t (n).

Characteristics:

1. With 0 as the center, the symmetrical single-peak distribution of the left and right; 2. The T-distribution is a cluster of curves whose morphological changes are related to the size of N (exactly, the degree of freedom ν). The smaller the Freedom ν, the lower the t distribution curve, and the greater the freedom ν, the more the T distribution curve is closer to the standard normal distribution (U distribution) curve. When the degree of freedom is large enough, the T distribution approximates the N (0,1) distribution. The upper ª of the T distribution: F-Distribution: u~x2 (N1), v~x2 (N2), and U,v Independent, the random variable is said to obey the F-distribution of degrees of freedom (N1,N2). Recorded as F~f (N1,N2).

The upper ª of the F-distribution:

Distribution of sample mean and sample variance in normal population

Probability theory and mathematical Statistics-ch6-sample and sample distribution