이공계용 확률과 통계

∙ 저자 서문 / iii

제1장 통계학 개요 ································································································ 1
1.1 · 개요 ·············································································································1
1.2 · 데이터 수집과 기술통계학 ········································································1
1.3 · 추론통계학과 확률모형 ··············································································2
1.4 · 모집단과 표본 ····························································································3
1.5 · 통계학의 역사 ····························································································3
연습문제 ·········································································································8

제2장 기술통계학 ······························································································· 11
2.1 · 서론 ···········································································································11
2.2 · 데이터 집합의 기술 ·················································································12
2.2.1 · 빈도 표와 빈도 그래프 ·······································································12
2.2.2 · 상대빈도 표와 상대빈도 그래프 ························································14
2.2.3 , , , - · 그룹 데이터 히스토그램 누적빈도그래프 줄기 잎 도표 ··············16
2.3 · 데이터 집합에 대한 요약 ········································································19
2.3.1 , , · 표본평균 표본중앙값 표본최빈값 ·····················································20
2.3.2 · 표본분산과 표본표준편차 ····································································24
2.3.3 · 표본 백분위수와 상자도표 ··································································26
2.4 · 체비셰프 부등식 ·······················································································29
2.5 · 정규 데이터 집합 ·····················································································33
2.6 · 쌍체 데이터 집합과 표본상관계수 ·························································36
2.7 · 로렌츠 곡선과 지니 지수 ········································································43
2.8 R · 사용하기 ·······························································································49
연습문제 ·······································································································53

제3장 확률론······································································································ 65
3.1 · 개요 ···········································································································65
3.2 · 표본공간과 사건 ·······················································································66
3.3 · 벤 다이어그램과 사건들에 대한 대수 법칙 ··········································68
3.4 · 확률에 대한 공리 ·····················································································69
3.5 · 균등 확률 결과들로 이루어지는 표본공간 ············································72
3.6 · 조건부 확률 ······························································································77
3.7 · 베이즈 공식 ······························································································81
3.8 · 독립사건 ····································································································88
연습문제 ·······································································································92

제4장 확률변수와 기댓값 ················································································· 103
4.1 · 확률변수 ································································································103
4.2 · 확률변수의 유형 ····················································································106
4.3 · 결합분포를 따르는 확률변수 ································································109
4.3.1 · 서로 독립인 확률변수 ·······································································115
4.3.2 · 조건부 분포 ·······················································································118
4.4 · 기댓값 ·····································································································121
4.5 · 기댓값의 성질 ························································································125
4.5.1 · 확률변수의 합에 대한 기댓값 ··························································128
4.6 · 분산 ·········································································································132
4.7 · 공분산과 확률변수의 합에 대한 분산 ·················································137
4.8 · 적률생성함수 ··························································································142
4.9 · 체비셰프 부등식과 대수의 약법칙 ·······················································143
연습문제 ·····································································································147

제5장 확률변수와 분포 ···················································································· 159
5.1 · 베르누이 확률변수와 이항 확률변수 ···················································159
5.1.1 R · 을 이용한 이항 분포함수의 계산 ··················································165
5.2 · 푸아송 확률변수 ····················································································166
5.2.1 R · 을 이용한 푸아송확률의 계산 ·······················································174
5.3 · 초기하 확률변수 ····················································································175
5.4 · 균일 확률변수 ························································································179
5.5 · 정규 확률변수 ························································································187
5.6 · 지수 확률변수 ························································································198
5.6.1 · 푸아송 과정 ·······················································································201
5.6.2 · 파레토 분포 ·······················································································204
5.7 · 감마분포 ·································································································207
5.8 · 정규분포의 파생 분포 ···········································································210
5.8.1 · 카이제곱분포 ······················································································210
5.8.2 t - 분포 ································································································214
5.8.3 F - 분포 ·······························································································217
5.9 · 로지스틱분포 ··························································································218
5.10 R · 에서의 분포 ·······················································································219
연습문제 ·····································································································221

제6장 표본 통계량의 분포 ··············································································· 229
6.1 · 개요 ·········································································································229
6.2 · 표본평균 ·································································································230
6.3 · 중심극한정리 ··························································································232
6.3.1 · 표본평균의 근사분포 ·········································································236
6.3.2 · 얼마나 큰 표본이 필요할까 ···························································238
6.4 · 표본분산 ·································································································238
6.5 · 정규 모집단에서의 표본분포 ································································239
6.5.1 · 표본평균의 분포 ················································································240
6.5.2 x와 S2의 결합분포 ··········································································240
6.6 · 유한 모집단에서의 표본추출 ································································243
연습문제 ·····································································································247

제7장 모수 추정 ······························································································· 253
7.1 · 개요 ·········································································································253
7.2 · 최대우도 추정량 ····················································································254
7.2.1 · 수명분포에 대한 추정 ·······································································262
7.3 · 구간추정치 ······························································································264
7.3.1 · 분산을 모를 때 정규 평균에 대한 신뢰구간 ·································270
7.3.2 · 예측구간 ·····························································································275
7.3.3 · 정규분포의 분산에 대한 신뢰구간 ··················································276
7.4 · 두 정규 모집단의 평균의 차에 대한 추정 ·········································278
7.5 · 베르누이 확률변수의 평균에 대한 근사 신뢰구간 ····························283
7.6 · 지수분포의 평균에 대한 신뢰구간 ·······················································287
7.7 · 점추정량에 대한 평가 ···········································································288
7.8 · 베이즈 추정량 ························································································295
연습문제 ·····································································································301

제8장 가설검정 ································································································· 313
8.1 · 개요 ·········································································································313
8.2 · 유의수준 ·································································································313
8.3 · 정규 모집단의 평균에 대한 검정 ························································315
8.3.1 · 분산을 알고 있는 경우 ·····································································315
8.3.2 : 분산을 모를 때 t - 검정 ··································································327
8.4 · 두 정규 모집단의 평균의 동일성에 대한 검정 ··································334
8.4.1 · 분산을 알고 있을 때 ········································································334
8.4.2 · 분산을 모를 때 ··················································································336
8.4.3 · 분산을 모르고 또 다를 때 ·······························································340
8.4.4 쌍체 t - 검정 ·······················································································341
8.5 · 정규 모집단의 분산에 관한 가설검정 ·················································343
8.5.1 · 두 정규 모집단의 분산의 동일성에 대한 검정 ······························344
8.6 · 베르누이 모집단에 대한 가설검정 ·······················································346
8.6.1 · 두 베르누이 모집단에서 모수 동일성에 대한 검정 ······················349
8.7 · 푸아송분포의 평균에 관한 검정 ··························································352
8.7.1 · 두 푸아송 모수 사이의 관계에 대한 검정 ·····································353
연습문제 ·····································································································356

제9장 회귀 ········································································································ 373
9.1 · 개요 ·········································································································373
9.2 · 회귀계수의 최소제곱 추정량 ································································374
9.3 · 추정량의 분포 ························································································378
x 이공계용 확률과 통계
9.4 · 회귀계수에 관한 통계적 추론 ······························································385
9.4.1 B에 관한 추론 ····················································································385
9.4.2 a에 대한 추론 ···················································································393
9.4.3 평균 반응치 a + Bx에 대한 추론 ····················································394
9.4.4 · 미래 반응치에 대한 예측구간 ··························································396
9.4.5 · 분포 결과들의 요약 ··········································································399
9.5 · 결정계수와 표본상관계수 ······································································399
9.6 : · 잔차의 분석 모형 평가 ·······································································402
9.7 · 선형성 변환 ····························································································404
9.8 · 가중최소제곱법 ······················································································407
9.9 · 다항식 회귀 ····························································································413
9.10 · 다중선형회귀 ························································································416
9.10.1 · 미래 반응치의 예측 ········································································426
9.10.2 · 범주형 데이터를 위한 가변수 ·······················································430
9.11 · 이진 출력 데이터를 위한 로지스틱 회귀모형 ··································432
연습문제 ·····································································································436

제10장 분산분석 ································································································· 459
10.1 · 서론 ······································································································459
10.2 · 개요 ······································································································460
10.3 · 일원 분산분석 ······················································································462
10.3.1 R · 을 이용한 계산 ·············································································469
10.3.2 · 표본평균들에 대한 다중 비교 ·······················································472
10.3.3 · 표본의 크기가 다를 때의 일원 분산분석 ·····································474
10.4 : · 이원 분산분석 원리와 모수의 추정 ················································476
10.5 : · 이원 분산분석 가설검정 ···································································479
10.6 · 상호작용이 있는 경우의 이원 분산분석 ···········································485
연습문제 ·····································································································493

제11장 적합도검정과 범주형 데이터 분석 ························································ 505
11.1 · 개요 ······································································································505
11.2 · 모든 모수가 정해졌을 때의 적합도검정 ···········································506
11.2.1 · 시뮬레이션에 의한 기각역의 결정 ················································511
11.3 · 일부 모수가 정해져 있지 않을 때의 적합도검정 ····························514
11.4 · 분할표에서의 독립성 검정 ·································································516
11.5 · 고정 주변합을 갖는 분할표에서의 독립성 검정 ······························520
11.6 - · 연속 데이터에 대한 콜모고로프 스미르노프 적합도검정 ···············523
연습문제 ·····································································································528

제12장 비모수적 가설검정 ················································································· 535
12.1 · 개요 ······································································································535
12.2 · 부호검정 ·······························································································535
12.3 · 부호순위검정 ························································································539
12.4 2 · 표본 문제 ···························································································544
12.4.1 · 다중 확률분포의 동일성에 대한 검정 ··········································547
12.5 · 무작위성에 대한 런 검정 ···································································550
연습문제 ·····································································································554

제13장 품질관리 ································································································· 561
13.1 · 개요 ······································································································561
13.2 : 평균값을 위한 관리도
- 관리도 ··················································562
13.2.1 u와 a를 모를 경우 ·········································································565
13.3 S - 관리도 ·····························································································570
13.4 · 불량률에 대한 관리도 ·········································································573
13.5 · 불량 개수에 대한 관리도 ···································································575
13.6 · 모집단평균의 변화를 탐지하기 위한 관리도 ····································579
13.6.1 · 이동평균 관리도 ··············································································579
13.6.2 · 지수 가중 이동평균 관리도 ···························································582
13.6.3 · 누적합 관리도 ·················································································587
연습문제 ·····································································································590

제14장 수명검사 ································································································· 597
14.1 · 개요 ······································································································597
14.2 · 위험률 함수 ·························································································597
14.3 · 수명검사에서의 지수분포 ···································································600
14.3.1 - 동시검사 r 번째 고장에서 종료 ····················································600
14.3.2 · 순차검사 ···························································································606
14.3.3 - 고정된 시각에 종료 · 동시검사 ·····················································610
14.3.4 · 베이지안 방식 ·················································································612
14.4 2 · 표본 문제 ···························································································614
14.5 · 수명검사에서의 와이블분포 ································································616
14.5.1 · 최소제곱법에 의한 모수 추정 ·······················································618
연습문제 ·····································································································621

제15장 시뮬레이션 부트스트랩 방법 순열 검정 , , ············································ 627
15.1 · 개요 ······································································································627
15.2 · 난수 ······································································································627
15.2.1 · 몬테카를로 시뮬레이션 방식 ·························································630
15.3 · 부트스트랩 방법 ··················································································631
15.4 · 순열 검정 ·····························································································639
15.4.1 · 순열 검정에서의 정규 근사 ···························································642
15.4.2 2 · 표본 순열 검정 ·············································································645
15.5 · 이산확률변수의 생성 ···········································································647
15.6 · 연속확률변수의 생성 ···········································································649
15.6.1 · 정규 확률변수의 생성 ····································································651
15.7 · 몬테카를로 실험에서 시뮬레이션 시행 횟수의 결정 ·······················652
연습문제 ·····································································································654

제16장 기계학습과 빅데이터 ············································································· 657
16.1 · 개요 ······································································································657
16.2 · 비행지연 확률 ······················································································658
16.3 · 일반 베이즈 접근 ················································································659
16.3.1 · 일반 베이즈 접근의 변형 ·······························································662
16.4 . k - 근접이웃기법 · 거리기반 탐지 ······················································665
16.4.1 · 거리 가중치기법 ··············································································666
16.4.2 · 구성요소 가중치 거리 ····································································667
16.5 · 접근법에 대한 평가 ············································································668
16.6 · 계량 가능한 특성화 벡터 ···································································670
16.6.1 · 근접 이웃 기법 ················································································670
16.6.2 · 로지스틱 회귀 ·················································································671
16.7 : 벤딧 문제 · 최고의 확률 선택 ···························································672
연습문제 ·····································································································675

부록 도표 ········································································································ 677
A1 · 표준정규 분포함수 ·················································································677
A2 ANOVA · 다중비교 상수 ·········································································678
∙ 색 인 / 67