Sample distribution - Hyohonbunpu

Japanese: 標本分布 - ひょうほんぶんぷ

When making statistical inference, we often take an arbitrary sample of size n from a population, consider a function of the sample, such as the arithmetic mean of the sample, and consider what probability distribution it has. In other words, we assume that n random variables X ₁ , X ₂ , …, X _n are independent and each X _i has the same probability distribution, and the problem is to determine the probability distribution of the random variable Y=F(X ₁ , …, X _n ), where F(x ₁ , …, x _n ) is a function of the given n variables. The distribution of Y is called the sample distribution.

In what follows, we assume that X ₁ , …, X _n are independent and that the distribution of each X _i is normal.

(1) If all X _i have the same normal distribution N(m,σ ² ), then

The distribution of is normal distribution N(m,σ ² /n).

(2) If all X _i have the same normal distribution N(0,1), then Y = X ₁ ² + X ₂ ² + ... + X _n ²
If the probability density of is k _n (x),

Here, Γ(x) represents the gamma function. The probability distribution with this k _n (x) as the probability density is called the χ ² (chi-squared) distribution with n degrees of freedom.

(3) If all X _i have the same normal distribution N(m,σ ² ), then

are independent, and the distribution of (n/σ ² )S ² is a χ ² distribution with (n-1) degrees of freedom.

(4) If X and _Y are independent, X has a normal distribution N(0,1) and Y has a ^χ2 distribution with n degrees of freedom, then

The probability density of

This distribution of Z _n is called the t-distribution with n degrees of freedom.

(5) If X _m and Y _n are independent and have χ ² distributions with m and n degrees of freedom, respectively,

The probability density f _mn (x) is

This probability distribution of Z is called the F distribution with degrees of freedom m and n.

[Shigeru Furuya]

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

統計的推測を行う場合、母集団から大きさnの任意標本をとり、たとえばその標本の相加平均など、標本のある関数を考え、これがどのような確率分布をもつかが問題となることが多い。すなわち、n個の確率変数X₁、X₂、……、X_nが独立で、各X_iは同一の確率分布をもつとし、F(x₁,……,x_n)を与えられたn変数の関数として、確率変数Y=F(X₁,……,X_n)の確率分布を決定することが問題となる。Yの分布のことを標本分布という。

　以下では、X₁、……、X_nは独立で、各X_iの分布は正規分布であると仮定する。

（1）X_iがすべて同一の正規分布N(m,σ²)をもてば

の分布は正規分布N(m,σ²/n)である。

（2）X_iがすべて同一の正規分布N(0,1)をもてば
　　Y=X₁²+X₂²+……+X_n²
の確率密度をk_n(x)とすると

である。ただしΓ(x)はガンマ関数を表す。このk_n(x)を確率密度とする確率分布を自由度nのχ²（カイ二乗）分布という。