RIO - Text.p65

Îãëàâëåíèå

ÎÃËÀÂËÅÍÈÅ

Ïðåäèñëîâèå .......................................................................................... 6

Ãëàâà 1. Îñíîâíûå ïîíÿòèÿ òåîðèè ìíîæåñòâ

è òåîðèè âåðîÿòíîñòåé ...................................................................... 11

1.1. Ìíîæåñòâà è ôóíêöèè ................................................................... 11

1.2. Ïðîñòðàíñòâà ýëåìåíòàðíûõ ñîáûòèé è ñëó÷àéíûå âåëè÷èíû.

Íåçàâèñèìîñòü ñîáûòèé è ñëó÷àéíûõ âåëè÷èí ................................. 18

1.3. Îæèäàåìîå çíà÷åíèå è äèñïåðñèÿ ñëó÷àéíîé âåëè÷èíû.

Êîâàðèàöèÿ è êîððåëÿöèÿ ñëó÷àéíûõ âåëè÷èí ................................... 29

1.4. Ôóíêöèè ðàñïðåäåëåíèÿ è ôóíêöèè ïëîòíîñòè ñëó÷àéíûõ âåëè÷èí .............................................................................. 44

1.5. Óñëîâíûå âåðîÿòíîñòè. Ôîðìóëà ïîëíîé âåðîÿòíîñòè.

Ôîðìóëà Áàéåñà ................................................................................... 53

1.6. Çàäà÷à êàâàëåðà äå Ìåðå ............................................................. 62

Ãëàâà 2. Ýëåìåíòû êîìáèíàòîðíîãî àíàëèçà ............................... 64

2.1. Âûáîðêè ñ âîçâðàùåíèåì è áåç âîçâðàùåíèÿ, ïåðåñòàíîâêè è êîìáèíàöèè ................................................................ 65

2.2. Òðåóãîëüíèê Ïàñêàëÿ è áèíîì Íüþòîíà ..................................... 68

2.3. Ôîðìóëà Áåðíóëëè ........................................................................ 70

2.4. Òåîðåìû Ìóàâðà — Ëàïëàñà ....................................................... 75

Ãëàâà 3. Íîðìàëüíîå ðàñïðåäåëåíèå............................................... 79

3.1. Îïðåäåëåíèå íîðìàëüíîãî ðàñïðåäåëåíèÿ ................................ 79

3.2. Öåíòðàëüíàÿ ïðåäåëüíàÿ òåîðåìà. Ðàñïðåäåëåíèå

âûáîðî÷íîãî ñðåäíåãî è âûáîðî÷íîãî îòíîøåíèÿ ......................... 85

3.3. Ïðèìåðû èñïîëüçîâàíèÿ íîðìàëüíîãî ðàñïðåäåëåíèÿ

äëÿ ïðîâåðêè ãèïîòåç .......................................................................... 94

1

Îãëàâëåíèå

3.4. Ïðèìåðû èñïîëüçîâàíèÿ íîðìàëüíîãî ðàñïðåäåëåíèÿ

äëÿ ïðèíÿòèÿ ðåøåíèé. Îøèáêè ïåðâîãî è âòîðîãî ðîäà .............. 106

Ãëàâà 4. Ñòàòèñòè÷åñêèå îöåíêè ................................................... 113

4.1. Íåñìåùåííîñòü, ñîñòîÿòåëüíîñòü è ýôôåêòèâíîñòü îöåíîê ..................................................................................................113

4.2. Ñîîòíîøåíèå ìåæäó ïîãðåøíîñòüþ, ðèñêîì

è ðàçìåðîì âûáîðêè. Äîâåðèòåëüíûå èíòåðâàëû ............................119

Ãëàâà 5. χ2-ðàñïðåäåëåíèå ............................................................... 127

5.1. Îïðåäåëåíèå χ2-ðàñïðåäåëåíèÿ ............................................ 127

5.2. Ïðîâåðêà ãèïîòåç î ñîîòâåòñòâèè íàáëþäåíèé ïðåäïîëàãàåìîìó ðàñïðåäåëåíèþ âåðîÿòíîñòåé

ñëó÷àéíîé âåëè÷èíû .......................................................................... 130

5.3. Ïðîâåðêà ãèïîòåç î íåçàâèñèìîñòè ïðèçíàêîâ è ãèïîòåç îá îäíîðîäíîñòè .............................................................. 138

Ãëàâà 6. t-ðàñïðåäåëåíèå ................................................................. 144

6.1. Îïðåäåëåíèå t-ðàñïðåäåëåíèÿ ................................................... 144

6.2. Ïðîâåðêà ãèïîòåç î ñðåäíåì çíà÷åíèè íîðìàëüíî ðàñïðåäåëåííîé ñëó÷àéíîé âåëè÷èíû ñ íåèçâåñòíîé äèñïåðñèåé ......................................................................................... 149

6.3. Ïðîâåðêà ãèïîòåç î ðàâåíñòâå ñðåäíèõ çíà÷åíèé äâóõ íîðìàëüíî ðàñïðåäåëåííûõ ñëó÷àéíûõ âåëè÷èí .................... 154

Ãëàâà 7. F-ðàñïðåäåëåíèå ................................................................ 159

7.1. Îïðåäåëåíèå F-ðàñïðåäåëåíèÿ ................................................. 159

7.2. Ïðîâåðêà ãèïîòåç î ðàâåíñòâå äèñïåðñèé äâóõ íîðìàëüíî ðàñïðåäåëåííûõ ñëó÷àéíûõ âåëè÷èí ............................ 162

7.3. Ýëåìåíòû äèñïåðñèîííîãî àíàëèçà .......................................... 164

Ãëàâà 8. Ðàñïðåäåëåíèå Ïóàññîíà .................................................. 169

8.1. Îïðåäåëåíèå ðàñïðåäåëåíèÿ Ïóàññîíà .................................... 169

8.2. Ïóàññîíîâñêèé ïîòîê ñîáûòèé ................................................. 172

2

Îãëàâëåíèå

8.3. Ýêñïîíåíöèàëüíîå ðàñïðåäåëåíèå è âðåìÿ ìåæäó ïîÿâëåíèåì äâóõ ïîñëåäîâàòåëüíûõ ñîáûòèé

â ïóàññîíîâñêîì ïîòîêå ................................................................... 177

8.4. Àíàëèç ïðîñòåéøåé ñèñòåìû ìàññîâîãî îáñëóæèâàíèÿ ......... 179

8.5. Ñâÿçü ðàñïðåäåëåíèÿ Ïóàññîíà ñ èñïûòàíèÿìè Áåðíóëëè è ñ íîðìàëüíûì ðàñïðåäåëåíèåì ..................................................... 184

8.6. Îöåíêà ïàðàìåòðà ïóàññîíîâñêîãî ðàñïðåäåëåíèÿ ................. 188

Ãëàâà 9. Ïðîñòàÿ ðåãðåññèÿ ............................................................ 191

9.1. Êîððåëÿöèÿ äèíàìè÷åñêèõ ðÿäîâ .............................................. 192

9.2. Ïðîñòàÿ ðåãðåññèÿ äèíàìè÷åñêèõ ðÿäîâ ................................. 196

9.3. Ïðîâåðêà çíà÷èìîñòè ïðîñòîé ðåãðåññèè ................................ 201

Ãëàâà 10. Ìíîæåñòâåííàÿ ðåãðåññèÿ ............................................ 207

10.1. Ìíîæåñòâåííàÿ ðåãðåññèÿ äèíàìè÷åñêèõ ðÿäîâ .................. 208

10.2. Ïðîâåðêà çíà÷èìîñòè îòäåëüíûõ ôàêòîðîâ è çíà÷èìîñòè ìíîæåñòâåííîé ðåãðåññèè â öåëîì .......................... 210

10.3. Àëãåáðàè÷åñêèå ïðåîáðàçîâàíèÿ è ïðåîáðàçîâàíèÿ

òèïà ñäâèãîâ è ðàçíîñòåé. Ôèêòèâíûå ïåðåìåííûå ........................ 217