Sixth chapter statistical quantity and its sampling distribution _ statistic

1 Statistic Quantity

Statistics: Constructs a function from a sample, without relying on any parameters
Common statistics: Sample mean (X¯), sample variance (S 2), sample variation coefficient (V=sx¯), sample K-Order distance, sample K-Order center distance, sample skewness, sample kurtosis
Order Statistics: Sample extreme difference (maximum minus minimum value)
Sufficient statistics: Statistics are not lost in the processing of statistical data. Discriminant theorem: Factorization theorem 2 distribution of several important distributions derived from normal distribution 1χ2

Set random variable x 1, x 2, ..., x n is independent of each other, and X i (i−1,2,..., N) obeys the standard normal distribution N (0,1), their squares and ∑n i=1 X 2 I obey the χ2 distribution of the degrees of Freedom N.
E (χ2) =n;d (χ2) =2n
The χ2 distribution is additive, where χ2 1∼χ2 (n 1), χ2 2∼χ2 (n 2), and independent, then
Χ1 1 +χ2 2∼χ2 (n 1) +χ2 (n 2)
The density function of the χ2 is shown in the following figure:
2 T distribution

Set random variable x∼n (0,1), y∼χ2 (N), and X and Y are independent, then
T=xy/n−−−−√
called the T distribution, which is recorded as T (n), where n is the degree of freedom.
When N≥2, the T-distribution of the mathematical expectation E (t) =0
When n≥3, T-distribution variance D (t) =nn−2
The density function of T is shown in the following figure:
3 F Distribution

Let the random variable y and z be independent of each other, and Y and Z obey the χ2 distributions of the DOF m and N respectively, and the random variable X has the following expression:
x=y/mz/n =nymz
It is said that X obeys the first degree of freedom as M, and the second degree of freedom is n distribution, which is written F (m,n)
E (X) =nn=2, n>2
D (X) =2n 2 (m+n−2) m (n−2 (n−4)), n>4
The density function of F is shown in the following figure:

If the random variable x obeys t (n), then x 2 obeys the F (1,n) distribution. This is useful in regression analysis of regression coefficient test. The distribution of 3 sample mean values in the central limit theorem

When the overall distribution is N (μ,σ 2 )  , the sampling distribution of x ¯   is still normal, the expectation is μ , and the variance is σ 2 /n . This shows that when the average mean value μ  is estimated by the sample mean x ¯  , there is no deviation (unbiased); When the n  is getting bigger, the x ¯   spreads less and less, that is, X  ¯   estimates mu  more and more accurate.
in practice, the distribution of the population is not always normal or approximate, at which point the distribution of the x ¯   will depend on the overall distribution. -Central limit theorem
Central limit theorem: Set from the mean value of μ , Variance is σ 2   (finite) of any one of the total sample size is n  sample, when the n  sufficiently large, the sample mean value x ¯& The sampling distributions of nbsp;  are approximate to the normal distribution with the mean value as μ  and the variance as σ 2 /n .
Note: What is when n is sufficiently large? The distinction between large and small samples is not based on the size of the sample. The statistical inference and problem analysis under the condition of fixed sample size, no matter how large the sample size is, is called small sample problem, but the statistical inference and problem analysis under the condition of sample size n->∞ is called large sample problem. In general statistics, N≥30 is a large sample, and n<30 is a small sample that has always been an empirical statement. 4 Sampling distribution of sample proportions

If the number of individuals with a feature in a sample with a sample size of n is x , the sample scale is represented by p : p=x/n 
can estimate the total proportional π  with the sample scale p .
When the n  is sufficiently large, the distribution of the p ^   can be approximated by a normal distribution, at which point the p ^   obeys the mean π , the variance is pi (1−π) n   The
P ^ ∼n (1−π) n )   Distribution of the difference in the average value of 52 sample π,π

Set x 1¯ is the mean value of a sample of n 1 that is independently pumped from the overall x 1∼n (μ1, σ2 1), and x 2¯ is independently extracted from the total x 1∼n (μ2, σ2 2) of a sample with a capacity of N 2, then Yes
E (X 1−x 2) =μ1−μ2
D (X 1−x 2) =σ2 1 n 1 +σ2 2 N 2
If the two population is a normal distribution, then X 1−x 2 is also a normal distribution; when n 1 and N 2 are larger, general requirements are n1≥30,n2≥30, and the sampling distribution of X 1−x 2 can be approximated by the normal distribution regardless of the overall distributions. 6 Distribution of variance distributions of 1 samples on sample variance

Set x 1, x 2,..., x N as a sample from the normal distribution, you can launch:
The distribution of the sample variance S 2 is based on a normal distribution with a population of N (μ,σ2):
(n−1) Distribution of variance ratio of 22 samples of S 2/σ2∼χ2 (n−1)

Set x 1 ,x 2 ,..., x n 1    to come from the normal population n (μ 1 ,σ 2 1  )   A sample, y 1 ,y 2 ,..., y n 1    is derived from the normal population n (μ 2  σ 2 2 )   A sample, and x i   and y i   are independent of each other, the
s 2 x /s& Nbsp;2 y σ 2 1 /σ 2 2  =s 2 x /σ 2 1  s 2 y /σ 2 2  ∼f (n 1 −1,n 2 −1)