If n independent random variables ξ? ,ξ? , ..., ξn, all obey the standard normal distribution, then the sum of squares of these n random variables obeying the standard normal distribution constitutes a new random variable, and its distribution law is called chi-square distribution.
In the section of sampling distribution theory, sampling from a normal population is equivalent to sampling n independent and identically distributed normal random variables ξ 1, ξ2, …, ξn at one time. These n random variables standardize the mean and variance of the population (i= 1 …, n), and obviously each of them obeys the standard normality.
Extended data:
If the random variable X obeys the normal distribution with a mathematical expectation of μ and a variance of σ 2, it is recorded as N(μ, σ 2). The expected value μ of probability density function with normal distribution determines its position, and its standard deviation σ determines its distribution amplitude. When μ = 0 and σ = 1, the normal distribution is standard normal distribution.
This event is almost impossible to happen in the experiment. It can be seen that the probability that X falls outside (μ-3σ, μ+3σ) is less than three thousandths. In practical problems, it is often thought that the corresponding event will not happen, and the interval (μ-3σ, μ+3σ) can basically be regarded as the actual possible value interval of random variable X.