Chapter 6

sample

If are independent r.v. having common distribution F, then we say they constitute sample or random sample from distribution F

같은 분포에서 왔으니 항상 평균과 분산이 같다

Sample Mean

Let are samples with . 분포는 노말일수도, 포아송 분포일수도 있다.

Sample Mean

Central Limit Theorem

Central Limit Theorem

Let be i.i.d. (independent, identically distributed) r.v. with mean , Variance Then, for large, distribution of is approximately

이것의 의의는 이 커지면 로 standard normalization이 가능하다는 것

위의 Sample Mean 하고 비교해보자 결론은 똑같다

Sample Variance

Sample Variance

The statistic Let s.t.

Sampling Distributions from Normal Population

Multivariate Normal Distribution

교수님의 추가 항목

Gaussian Random Vector (Normal Random Vector)

Let is Gaussian random vector, or Normal random vector have Multivariate normal distribution, or Multivariate Gaussian distribution, or Joint normal distribution If can be expressed as

여기서 이고 는 모두 i.i.d.

Proposition.

Let be Gaussian random vector. Then,

Proposition

be normal r.v.,

Proposition

: i.i.d. and

Thm 6.5.1.

: sample such that . Then,

3. This is Student’s Theorem

Cor 6.5.2.

Let sample from normal population. Then,

Sampling from Finite population

For any , as