( ) ( ) ( ) ( ) è ø. P A Wahrscheinlichkeitsmaß. lim n. Dr. Christian Schwarz 4. KOMBINATORIK Permutationen

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1 BBA Projektsemar Thess Dr. Chrsta Schwarz Formelsammlug Aalytsche Statstk 4. KOMBINATORIK 4.. Permutatoe Azahl der Permutatoe vo N Elemete ohe Wederholug: Multomalkoeffzet: N! = N N- N N! N! N!... N! k 4.. Kombatoe Zehe vo Elemete aus N Elemete Azahl möglcher Varatoe bzw. Kombatoe mt Aordug (Varato) ohe Aordug (Kombato) mt Wederholug (mt Zurücklege ) N ohe Wederholug (ohe Zurücklege ) N! N-! ( )! ænö N! ç =! ( N -)! ( N-) æn- + ö N- + ç = è ø è ø!! 5. WAHRSCHEINLICHKEITSBEGRIFF UND RECHENREGELN 5.. Notato W Ergebsmege ees Zufallsvorgags w Elemetareregs A, B Eregsse P A Wahrschelchketsmaß 5.. Wahrschelchkete Aome der Wahrschelchketstheore: Aom : P( A ) ³ für jedes Eregs A Aom : P ( W ) = Aom 3: P( A A A ) P( A ) P( A ) P( A ) Wahrschelchketsbegrff ach Laplace: P( A ) = = Wahrschelchketsbegrff ach Mses: f ( A) = P( A ) lm A W È È È... = für A Ç A = für alle = j j Azahl der für A güstge Fälle Azahl aller möglche Fälle (Zufallsvorgag wrd belebg oft wederholt: ) /

2 BBA Projektsemar Thess Dr. Chrsta Schwarz Formelsammlug Aalytsche Statstk 5.. Sätze der Wahrschelchketsrechug Addtossatz: P( AÈ B) = P( A) + P( B) -P( AÇB ) Bedgte Wahrschelchket: P( A B) ( ÇB) PB P A = mt: P B > A ud B sd uabhägg, falls glt: P( A B) = P( A) bzw. P( B A) = P( B ) Multplkatossatz: P( AÇ B) = P( A B) P( B) = P( B A) P( A ) P( AÇ B) = P( A) P( B) ÛA ud B uabhägg 5.3. Totale Wahrschelchket ud Bayes-Theorem Notato A, A,... A Eregsse mt A ÈA È... È A =W B Eregs mt bekate bedgte W kete P B A ; = ;...; Formel Totale Wahrschelchket: PB = PBA ( ) PA Satz vo Bayes: P( Aj B) å = = ( ) PA PBA j PAj PBA j PAj = = PB PBA å 7. ZUFALLSVARIABLE UND VERTEILUNGSFUNKTION 7.. Notato X Zufallsvarable (dskret oder stetg) Realsato vo X f() Wahrschelchketsfukto (dskrete ZV) oder Dchte (stetge ZV) F = P X Vertelugsfukto 7.. Dskrete Zufallsvarable 7... Fuktoe ud Maßzahle dskreter Zufallsvarable Wahrschelchketsfukto: f( ) P( X ) ìp falls = = ;;3;... = = =í î sost /

3 BBA Projektsemar Thess Dr. Chrsta Schwarz Formelsammlug Aalytsche Statstk Vertelugsfukto F( ) P( X ) f( ) = =å å Erwartugswert: E( X) = f( ) Varaz: Var ( X ) = é - E( X ) ù f ( ) Stadardabwechug: s X = Var( X ) åë û 7... Wchtge dskrete Wahrschelchketsverteluge æ = = = ö f P X ç p - p ; =,,,..., è ø Bomalvertelug B(;p): - mt: E( X) = p Hypergeometrsche Vertelug H(N;M;): f( ) P( X ) = ( - ) Var X p p æmö æn-mö ç ç - = = = è ø è ø ænö ç è ø mt: = M E X N M N-M N- Var ( X) = N N N - l = = = ; l >! -l Posso-Vertelug P ( l ) : f( ) P( X ) e f p ;,..., Dskrete Glechvertelug: = = = 7.3. Stetge Zufallsvarable Fuktoe ud Maßzahle stetger Zufallsvarable Egeschafte der Dchtefukto: () f( ) ³ () ò f d = - (3) ò b f ( ) d = P ( a X b ) a 3/

4 BBA Projektsemar Thess Dr. Chrsta Schwarz Formelsammlug Aalytsche Statstk Vertelugsfukto: F = P X =ò f t dt Allgemee Gesetze: Pa ( X b) = Fb -Fa P( X a) = F( a) P( X ³ a) = -F( a) + Erwartugswert: E( X) = f( ) d( ) + Varaz: Var ( X ) = é - E( X ) ù f ( ) d( ) Stadardabwechug: s X = Var( X ) ò - ò - ë - û Wchtge stetge Wahrschelchketsverteluge A. Stetge Glechvertelug Dchte der stetge Glechvertelug: ì ï f = íb- a ï î für a b sost Vertelugsfukto der Glechvertelug: ì falls < a ï - a F = í falls a b ïb - a ïî falls > b a+ b = Erwartugswert ud Varaz: E( X ) Var ( X ) = ( b- a) B. Normalvertelug Dchte der N(µ;s)-vertelte ZV: Vertelugsfukto der N(µ;s)-vertelte ZV: f = e s p æ-µ ö - ç è s ø æt-µ ö - ç è s ø F = s p ò e dt Erwartugswert ud Varaz: E( X ) = µ Var ( X ) = s - 4/

5 BBA Projektsemar Thess Dr. Chrsta Schwarz Formelsammlug Aalytsche Statstk Stadardserug der N(µ;s)-vertelte X - µ s Y = Þ Y! N Zufallsvarable X: ( ;) æ - µ ö = = ç è s ø mt: F( ) P( X ) Φ [ tabellert] 7.4. Appromatoe Appromato der Bomalvertelug: () Für ³ 5; p, ud p ; ud p st ee B(; p)-vertelte ZV äherugswese P(p)- vertelt. p - p > 9 st ee B(; p)-vertelte ZV () Für äherugswese Np p( p ) ; - -vertelt. Appromato der hypergeometrsche Vertelug: Für N st ee H(N; M; )-vertelte ZV æ M ö äherugswese Bç ; - vertelt. è N ø Appromato der Posso-Vertelug: Für l > st ee P( l ) äherugswese N -vertelte ZV l; l - vertelt. 8. PUNKT-SCHÄTZUNG Ubekater Parameter der Grudgesamthet: Schätzfukto: Mttelwert µ : = = å Mttelwert der Stchprobewerte Varaz s : s = å( -) - = Varaz der Stchprobewerte Atelswert p: p = k Atel der Stchprobe 8. INTERVALL-SCHÄTZUNG 9.. Itervall-Schätzug: Kofdeztervall für µ be ormalvertelter Grudgesamthet ud bekater Varaz Schrtt : E Kofdezveau ( - a) wrd festgelegt (z.b.,95 oder,99). Schrtt : Der für das gewählte Kofdezveau zugehörge Wert c wrd ermttelt, z.b. ( - a),9,95,99,999 c,65,96,58 3,9 5/

6 BBA Projektsemar Thess Dr. Chrsta Schwarz Formelsammlug Aalytsche Statstk Schrtt 3: Schrtt 4: Der Stchprobemttelwert wrd ermttelt. Ma berechet das Vertrauestervall s s - c µ + c å. - Falls s ubekat ud > 3: Appromato vo s durch s mt s= ( -) = 9.. Kofdeztervall für de Atelswert p Schrtt : E Kofdezveau ( - a) wrd festgelegt (z.b.,95 oder,99). Schrtt : Der zu dem gewählte Kofdezveau zugehörge Wert c wrd ermttelt. ( - a):,9,95,99,999 c,65,96,58 3,9 Schrtt 3: Der Atel p = k der Stchprobe wrd berechet. Schrtt 4: Falls p - p > 9 glt, so lautet das Kofdeztervall: p ( - p) p ( -p) p- c p p+ c p - p > 9 aufgrud kleer Stchprobe cht erfüllt st, so st de gezegte [Falls Vorgeheswese cht möglch, da ma cht mt der Normalvertelug arbete ka.]. ZWEISEITIGER TEST FÜR DEN MITTELWERT µ EINER NORMALVERTEILUNG Schrtt : Schrtt : Schrtt 3: Wahl eer Sgfkazzahl a ud Etahme des zugehörge c-wertes für ( - a). Berechug der Aahmegreze s cu = µ - c ud s co = µ + c Berechug des Mttelwerts der Stchprobe: Fällt de Aahmeberech, d.h. glt cu c, ehme ma de Hypothese o H: µ = µ a. Fällt de Ablehugsberech, lehe ma de Hypothese H: µ = µ ab. Bem esetge Test lautet de Hypothese H: µ ³ µ bzw. H: µ µ ud de Alteratvhypothese H : µ < µ bzw. H : µ > µ. Folglch st auch ur c u oder c o zu bereche. 6/

7 BBA Projektsemar Thess Dr. Chrsta Schwarz Formelsammlug Aalytsche Statstk BINOMIALKOEFFIZIENT: æ k ç ö èk ø /

8 BBA Projektsemar Thess Dr. Chrsta Schwarz Formelsammlug Aalytsche Statstk æ = ö f ç p - p ; ³ è ø WAHRSCHEINLICHKEITSFUNKTION DER B(N;P)-VERTEILUNG: - p,5,,5,,5,3,35,4,45,5,55,6,65,7,75,8,85,9,95,95,9,85,8,75,7,65,6,55,5,45,4,35,3,5,,5,,5,5,,5,,5,3,35,4,45,5,55,6,65,7,75,8,85,9,95,95,8,75,64,565,49,45,36,35,5,5,6,5,9,65,4,5,,5,95,8,55,3,375,4,455,48,495,5,495,48,455,4,375,3,55,8,95,5,,5,4,65,9,5,6,5,5,35,36,45,49,565,64,75,8,95 3,8574,79,64,5,49,343,746,6,664,5,9,64,49,7,56,8,34,, 3,354,43,35,384,49,44,4436,43,484,375,334,88,389,89,46,96,574,7,7 3,7,7,574,96,46,89,389,88,334,375,484,43,4436,44,49,384,35,43, ,,,34,8,56,7,49,64,9,5,664,6,746,343,49,5,64,79,8574 4,845,656,5,496,364,4,785,96,95,65,4,56,5,8,39,6,5,, 4,75,96,3685,496,49,46,3845,3456,995,5,5,536,5,756,469,56,5,36,5 4,35,486,975,536,9,646,35,3456,3675,375,3675,3456,35,646,9,536,975,486,35 4 3,5,36,5,56,469,756,5,536,5,5,995,3456,3845,46,49,496,3685,96,75 4 4,,,5,6,39,8,5,56,4,65,95,96,785,4,364,496,5,656,845 5,7738,595,4437,377,373,68,6,778,53,33,85,,53,4,,3,,, 5,36,38,395,496,3955,36,34,59,59,563,8,768,488,84,46,64,,5, 5,4,79,38,48,637,387,3364,3456,3369,35,757,34,8,33,879,5,44,8, 5 3,,8,44,5,879,33,8,34,757,35,3369,3456,3364,387,637,48,38,79,4 5 4,,5,,64,46,84,488,768,8,563,59,59,34,36,3955,496,395,38,36 5 5,,,,3,,4,53,,85,33,53,778,6,68,373,377,4437,595,7738 6,735,534,377,6,78,76,754,467,77,56,83,4,8,7,,,,, 6,3,3543,3993,393,356,35,437,866,359,938,69,369,5,,44,5,4,, 6,35,984,76,458,966,34,38,3,78,344,86,38,95,595,33,54,55,, 6 3,,46,45,89,38,85,355,765,33,35,33,765,355,85,38,89,45,46, 6 4,,,55,54,33,595,95,38,86,344,78,3,38,34,966,458,76,984,35 6 5,,,4,5,44,,5,369,69,938,359,866,437,35,356,393,3993,3543,3 6 6,,,,,,7,8,4,83,56,77,467,754,76,78,6,377,534,735 7,6983,4783,36,97,335,84,49,8,5,78,37,6,6,,,,,, 7,573,37,396,367,35,47,848,36,87,547,3,7,84,36,3,4,,, 7,46,4,97,753,35,377,985,63,4,64,7,774,466,5,5,43,,, 7 3,36,3,67,47,73,69,679,93,98,734,388,935,44,97,577,87,9,6, 7 4,,6,9,87,577,97,44,935,388,734,98,93,679,69,73,47,67,3,36 7 5,,,,43,5,5,466,774,7,64,4,63,985,377,35,753,97,4,46 7 6,,,,4,3,36,84,7,3,547,87,36,848,47,35,367,396,37, ,,,,,,,6,6,37,78,5,8,49,84,335,97,36,4783,6983 8,6634,435,75,678,,576,39,68,84,39,7,7,,,,,,, 8,793,386,3847,3355,67,977,373,896,548,33,64,79,33,,4,,,, 8,55,488,376,936,35,965,587,9,569,94,73,43,7,,38,,,, 8 3,54,33,839,468,76,54,786,787,568,88,79,39,88,467,3,9,6,4, 8 4,4,46,85,459,865,36,875,3,67,734,67,3,875,36,865,459,85,46,4 8 5,,4,6,9,3,467,88,39,79,88,568,787,786,54,76,468,839,33,54 8 6,,,,,38,,7,43,73,94,569,9,587,965,35,936,376,488,55 8 7,,,,,4,,33,79,64,33,548,896,373,977,67,3355,3847,386, ,,,,,,,,7,7,39,84,68,39,576,,678,75,435,6634 9,63,3874,36,34,75,44,7,,46,,8,3,,,,,,, 9,985,3874,3679,3,53,556,4,65,339,76,83,35,3,4,,,,, 9,69,7,597,3,33,668,6,6,,73,47,,98,39,,3,,, 9 3,77,446,69,76,336,668,76,58,9,64,6,743,44,,87,8,6,, 9 4,6,74,83,66,68,75,94,58,6,46,8,67,8,735,389,65,5,8, 9 5,,8,5,65,389,735,8,67,8,46,6,58,94,75,68,66,83,74,6 9 6,,,6,8,87,,44,743,6,64,9,58,76,668,336,76,69,446,77 9 7,,,,3,,39,98,,47,73,,6,6,668,33,3,597,7,69 9 8,,,,,,4,3,35,83,76,339,65,4,556,53,3,3679,3874, ,,,,,,,,3,8,,46,,7,44,75,34,36,3874,63,5987,3487,969,74,563,8,35,6,5,,3,,,,,,,,,35,3874,3474,684,877,,75,43,7,98,4,6,5,,,,,,,746,937,759,3,86,335,757,9,763,439,9,6,43,4,4,,,, 3,5,574,98,3,53,668,5,5,665,7,746,45,,9,3,8,,, 4,,,4,88,46,,377,58,384,5,596,5,689,368,6,55,,, 5,,5,85,64,584,9,536,7,34,46,34,7,536,9,584,64,85,5, 6,,,,55,6,368,689,5,596,5,384,58,377,,46,88,4,, 7,,,,8,3,9,,45,746,7,665,5,5,668,53,3,98,574,5 8,,,,,4,4,43,6,9,439,763,9,757,335,86,3,759,937,746 9,,,,,,,5,6,4,98,7,43,75,,877,684,3474,3874,35,,,,,,,,,3,,5,6,35,8,563,74,969,3487,5987 8/

9 BBA Projektsemar Thess Dr. Chrsta Schwarz Formelsammlug Aalytsche Statstk STANDARDNORMALVERTEILUNG: Φ ( Y), falls Y > ,*,5,54,58,5,56,599,539,579,539,5359,*,5398,5438,5478,557,5557,5596,5636,5675,574,5753,*,5793,583,587,59,5948,5987,66,664,63,64,3*,679,67,655,693,633,6368,646,6443,648,657,4*,6554,659,668,6664,67,6736,677,688,6844,6879,5*,695,695,6985,79,754,788,73,757,79,74,6*,757,79,734,7357,7389,74,7454,7486,757,7549,7*,758,76,764,7673,774,7734,7764,7794,783,785,8*,788,79,7939,7967,7995,83,85,878,86,833,9*,859,886,8,838,864,889,835,834,8365,8389,*,843,8438,846,8485,858,853,8554,8577,8599,86,*,8643,8665,8686,878,879,8749,877,879,88,883,*,8849,8869,8888,897,895,8944,896,898,8997,95,3*,93,949,966,98,999,95,93,947,96,977,4*,99,97,9,936,95,965,979,99,936,939,5*,933,9345,9357,937,938,9394,946,948,949,944,6*,945,9463,9474,9484,9495,955,955,955,9535,9545,7*,9554,9564,9573,958,959,9599,968,966,965,9633,8*,964,9649,9656,9664,967,9678,9686,9693,9699,976,9*,973,979,976,973,9738,9744,975,9756,976,9767,*,977,9778,9783,9788,9793,9798,983,988,98,987,*,98,986,983,9834,9838,984,9846,985,9854,9857,*,986,9864,9868,987,9875,9878,988,9884,9887,989,3*,9893,9896,9898,99,994,996,999,99,993,996,4*,998,99,99,995,997,999,993,993,9934,9936,5*,9938,994,994,9943,9945,9946,9948,9949,995,995,6*,9953,9955,9956,9957,9959,996,996,996,9963,9964,7*,9965,9966,9967,9968,9969,997,997,997,9973,9974,8*,9974,9975,9976,9977,9977,9978,9979,9979,998,998,9*,998,998,998,9983,9984,9984,9985,9985,9986,9986 3,*,9987,9987,9987,9988,9988,9989,9989,9989,999,999 9/

10 BBA Projektsemar Thess Dr. Chrsta Schwarz Formelsammlug Aalytsche Statstk STANDARDNORMALVERTEILUNG: Φ ( Y), falls Y < ,*,,,,,,,,3,3,3 -,9*,4,4,5,5,6,6,7,8,8,9 -,8*,9,,,,,3,3,4,5,6 -,7*,6,7,8,9,3,3,3,33,34,35 -,6*,36,37,38,39,4,4,43,44,45,47 -,5*,48,49,5,5,54,55,57,59,6,6 -,4*,64,66,68,69,7,73,75,78,8,8 -,3*,84,87,89,9,94,96,99,,4,7 -,*,,3,6,9,,5,9,3,36,39 -,*,43,46,5,54,58,6,66,7,74,79 -,*,83,88,9,97,,7,,7,,8 -,9*,33,39,44,5,56,6,68,74,8,87 -,8*,94,3,37,34,3,39,336,344,35,359 -,7*,367,375,384,39,4,49,48,47,436,446 -,6*,455,465,475,485,495,55,56,56,537,548 -,5*,559,57,58,594,66,68,63,643,655,668 -,4*,68,694,78,7,735,749,764,778,793,88 -,3*,83,838,853,869,885,9,98,934,95,968 -,*,985,3,,38,56,75,93,,3,5 -,*,7,9,,3,5,7,9,34,335,357 -,*,379,4,43,446,469,49,55,539,56,587 -,9*,6,635,66,685,7,736,76,788,84,84 -,8*,867,894,9,949,977,5,33,6,9,9 -,7*,48,77,6,36,66,96,37,358,389,4 -,6*,45,483,54,546,578,6,643,676,79,743 -,5*,776,8,843,877,9,946,98,35,35,385 -,4*,3,356,39,38,364,33,3336,337,349,3446 -,3*,3483,35,3557,3594,363,3669,377,3745,3783,38 -,*,3859,3897,3936,3974,43,45,49,49,468,47 -,*,447,486,435,4364,444,4443,4483,45,456,46,*,464,468,47,476,48,484,488,49,496,5 /