The variance problem of chi square and t distribution! Ace enter!

1.Let x=y1 2+y2 2+y3 2+.yn 2 where yn is independent and obeys n(0,1).

Then x obeys a chi-square distribution with n degrees of freedom.

Then d(x) = d(y1 2) + d(y2 2)+d(yn 2) because yn is independent.

2n because d(yn 2) = e(yn 4)-e(yn 2) = 3-1 = 2

where the fourth-order expectation of the standard normal distribution is 3 or e(y n)=(2n)!/(n!2 n) where y is a standard normal random variable n is an odd number, if n is an even number, e(y n)=0 either directly the algorithm is a step-by-step integration method.

Or you can calculate the variance of the chi-square distribution directly, which is very easy to calculate, because the chi-square distribution with degrees of freedom n is actually a gamma distribution with coefficients n 2, 1 2, and the nature of the gamma function makes it easy to calculate any order expectation of x by the following means:

The nth power expectation of x is the density function multiplied by x n integrals, and then you put x n into the density function, and you get n 2-1+n power of x, i.e. the coefficients change from n 2 to n 2+n, and you also put the gamma function below the fraction and 1 2 (n 2) outside the integral, and then add the required coefficients (so that the formula becomes a gamma distribution with coefficients n 2+n and 1 2, one integral to 1), and then divide by the coefficients you added, and finally all the coefficients outside the integral are your x n of expectations.

2.Let x obey the chi-square distribution of n(0,1)z with degrees of freedom n x and z independent then d(t)=e(t 2)-e(t) 2 where e(t) = e(x sqrt(z n)) = e(x) * e(1 sqrt(z n)) = 0

So d(t)=e(t 2)=e(x 2 (z n)) = e(x 2)*e(n z)=n*e(x 2)*e(1 z).

where e(x 2)=1 e(1 z)=1 (n-2) (calculated by density function Same as in question 1 The 1 2nd power expectation of the chi-square distribution can be easily calculated).

So d(t)=n (n-2).

That's right, the density function of the chi-square distribution with degrees of freedom k is.

You understand me better than this function.

Because he obeys a chi-square distribution, his variance is 2 times the degrees of freedom (the nature of the chi-square distribution, as evidenced in the probability book).

So d(x)=2n

e（x）＝n

Not really.

<> did it before, a truth.

Well, isn't xi 2 subject to a chi-square distribution with a degree of freedom of 1? Because the chi-square traveler trembled and the canvas was expected to be free to hail, the variance was 2*degrees of freedom. So d(xi 2)=2.

Chi-square distributionThe expectation and variance are: e(x)=n, d(x)=2n

t-distribution. e(x)=0(n>1)，d(x)=n/(n-2)(n>2)

f(m,n) distribution: e(x)=n (n-2)(n>2).

d(x)=[2n^2*(m+n-2)]/m(n-2)^2*(n-4)](n>4)

Chi-square distribution (2 distribution) is a probability distribution commonly used in probability theory and statistics, k independent standard normal distribution.

The sum of squares of the variables obeys degrees of freedom.

is the chi-square distribution of k, which is often used for hypothesis testing and confidence intervals.

calculations. The density function of the normal distribution is characterized by the fact that, with respect to symmetry, it reaches a maximum value at , and at positive (negative) infinity it takes a value of 0, and there is an inflection point at .

Its shape is destructive, with a neutral and low edge, and the image is a bell-shaped curve above the x-axis. When the residual beam 0, 2 1 is called, it is called the standard normal distribution and is denoted as n(0,1).

Binomial distribution: there are only two possible outcomes in each experiment, and the two outcomes are opposed to each other and independent of each other, and have nothing to do with the results of other experiments, and the probability of the event occurring or not remains unchanged in each independent trial, then this series of experiments is collectively called the n-fold Bernoulli experiment, and when the number of trials is 1, the binomial distribution obeys the 0-1 distribution.

t-distribution. It is used to test whether the mean is different. The f-distribution is used to test whether the variance is different. Chi-square distributionIt is mainly used to test whether the sample deviates from expectations, such as the distribution of deviations from expectations (goodness-of-fit test), the proportion of expectations (contingency table), etc.

t-test. The sum test can only use continuous data (quantitative data). Chi-square test.

You can use either continuous or discrete data (positive foci) or log-likelihood values. But the calculation formula is different.

t-distribution.

In probability theory and statistics, the student's t-distribution can be shortened to the t-distribution, which is used to estimate the mean of a population with a normal distribution and unknown variance based on a small sample. If the population variance is known (e.g., when the sample size is large enough), the population mean should be estimated using a normal distribution.

Suppose x is a normally distributed independent random variable (the expected value of the random variable is ?The variance is 2 but unknown). Order:

is the sample mean.

is the sample variance.

Chi-square distribution

Chi-square distribution (chi-square distribution, distribution) is a type of probability distribution commonly used in probability theory and statistics. If k random variables z1、......zks are independent of each other and conform to the standard normal distribution.

(Mathematical Expectation) of random variables.

is 0 and the variance is 1), then the sum of the squares of the random variable z.

It is called the degree of freedom of obedience.

is the chi-square distribution of k, denoted as.

f distribution

f distribution definition: Let x and y be two independent random variables, x obeys the chi-square distribution with degrees of freedom k1, y obeys the chi-square distribution of the free mountain and the degree k2, and the f- distribution is the distribution of the ratio of the two chi-square distribution variables x and y divided by their respective degrees of freedom:

The t-distribution is the basis for a student t-test for significance tests for the difference in the mean between the two samples, in the parent standard deviation.

In the case of unknown circumstances, the Student's t-test can be applied regardless of the size of the sample.

A chi-square distribution is a distribution of k independent standard normally distributed variables squared and obeyed by degrees of freedom in k, which can be used to calculate hypothesis testing and confidence intervals.

The Pearson chi-square test, which is extended by it, is commonly used.

The f distribution is based on the chi-square distribution.

The three well-known statistics of the t-distribution f distribution and the chi-square distribution based on the standard normal distribution variables are widely used in practice, because these three statistics not only have a clear background, but also have explicit expressions for the density function of the sampling distribution, which is called the "three major sampling distributions" in statistics.

These three sampling distributions are known as chi-square distribution, t-distribution and f-distribution.

t-distribution, f-square, and chi-square distribution

Before using the data, it is necessary to pay attention to the effective methods to collect data, such as designing a sampling plan, arranging experiments, etc. Only when data is collected effectively can data be used effectively for statistical inference. After obtaining the data of the town, the sampling distribution, i.e., the statistical model, was determined according to the characteristics of the problem and the sampling method.

Based on the statistical model, statistical inference problems can be performed in the following steps.

Seeking the exact distribution of statistics: When it is difficult to find the exact distribution of measurement, you can consider using the central limit theorem or other limit theorems to find the extreme ridge limit distribution of statistics.

Based on the exact distribution or limit distribution of the statistic, the exact solution or approximate solution of the statistical inference problem is obtained.

The second step is the most important, but also the most difficult. Statistics on the distribution of the three major distributions and the sample mean and sample variance under the normal population play an important role in finding the accurate distribution of statistics related to normal variables. This is especially evident when it comes to interval estimation and hypothesis testing.

x2 distribution,t-distribution. , f distribution, all three distributions are based onNormal distributionWhat is obtained by deformation can only be used in practiceHypothesis testing。For example, if we know that samples x are all samples that follow a normal distribution, and the variance is unknown, then the t-distribution will be used to test the uniformity of x.

x1,x2..xn all obey the normal distribution of n(0,1), then.

x1^2+x2^2+..Observe the x2(n) distribution.

It is equivalent to the formation of a new statistic y=x1 2+x2 2+.

Meaning of the parameter. The normal distribution has two parameters, i.e., expectation (mean) and standard deviation, with 2 being the variance.

A normal distribution has two parameters and 2 for the distribution of a continuous random variable, the first parameter is the mean of the random variable that obeys the normal distribution, and the second parameter 2 is the variance of this random variable, so the normal distribution is denoted as n( ,2).

is a positional parameter of the normal distribution, which describes the central trend position of the normal distribution. The probability law is that the probability of taking a value that is close to is high, and the probability of taking a value that is farther away is smaller. The normal distribution is axed with x= as the axis of symmetry.

The left and right sides are perfectly symmetrical. Expectation, mean, median of normal distribution.

If the mode is the same, it is equal to .

The chi-square distribution is the sum of squares of a number of numbers that obey a normal distribution, and a few numbers are chi-square distributions that obey a few degrees of freedom, and the t-distribution is that the numerator is a normal distribution, and the denominator is the root number below (the number of x-obeying is divided by the degrees of freedom n), which is the t-distribution that obeys the degrees of freedom n. Probability has just been completed.

Hope it helps.

Chi-square distributionThe period difference and variance are: e(x)=n, d(x)=2n.

t-distribution. e(x)=0(n>1)，d(x)=n/(n-2)(n>2)。

f(m,n) distribution: e(x)=n (n-2)(n>2).

d(x)=[2n^2*(m+n-2)]/m(n-2)^2*(n-4)](n>4)。

Brief introduction. We often refer to the number of independent variables in a formula as the "degrees of freedom" of the formula, and the way to determine the degrees of freedom of a formula is that if the formula contains n variable bonds, where k are restricted sample statistics, then the degrees of freedom of the expression are n-k.

For example, 1, 2 ,..., n, where 1- n-1 are independent of each other, and n is the average of the rest of the variables in the virtual light file.

Hence the degrees of freedom are n-1.

Theoretically, n independent and equally distributedRandom variables, all obey a normal distribution, then the distribution of the sum of squares obeys isDegrees of freedomfor nChi-square distribution

If n random variables are independent of each other 1, 2 ,..., n , all obey the standard normal distribution.

It is also called independent of the same distribution in the standard normal distribution), then the sum of the squares of the n random variables that obey the standard normal distribution i 2 constitutes a new random variable, and its chi-square distribution distribution law is called 2(n) distribution (chisquare distribution).

where the parameter n is called the degrees of freedom, and the difference in degrees of freedom is another 2 distribution, just as the difference in mean or variance in a normal distribution is another normal distribution.

Supplement: 2 are distributed in one quadrant.

internally, it is positively skewed.

With the increase of parameter n, the 2 distribution tends to be normal.

The mean of the 2 distribution is the degrees of freedom n, denoted as e 2=n, where the symbol "e" indicates the mean of the random variable; The variance of the 2 distribution is 2 degrees of freedom (2n), denoted as d 2=2n, where the symbol "d" indicates the variance for the random variable.

From the mean and variance of the 2 distribution, it can be seen that as the degree of freedom n increases, the 2 distribution extends in the positive infinity direction (because the mean n is getting larger and larger), and the distribution curve is getting lower and wider (because the variance 2n is getting larger and larger).

The 2 distribution is additive: if there are k random variables that obey the 2 distribution and are independent of each other, then the sum of them is still the 2 distribution, and the degrees of freedom of the new 2 distribution are the sum of the original k 2 distributions degrees of freedom. It is denoted as:

The 2 distribution is continuous, but some discrete distributions also obey the 2 distribution, which is especially extensive in the number of times.

(x1-2x2) obeys n(0,20), (3x3-4x4) obeys n(0,100), if y obeys the chi-square distribution, i.e. a*(x1-2x2) 2 obeys the standard normal distribution n(0,1), so a=1 20, in the same way, b=1 100, so y obeys chi-square (2), and the degrees of freedom are 2