Maple 16 - SandsynlighedsregningLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Vigtige redskaber i kombinatorik Vi skal se p\303\245 nogle grundelementer indenfor omr\303\245det Kombinatorik, som besk\303\246ftiger sig med at t\303\246lle antallet af mulige kombinationer i forskellige sammenh\303\246nge. Antal m\303\245der at permutere n elementerOfte har man brug for at vide, hvor mange forskellige r\303\246kkef\303\270lger, man kan stille n forskellige elementer op i. Det kunne fx v\303\246re de fire elementer 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. Svaret er :LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiNEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIiFGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHRkcvJStleGVjdXRhYmxlR0Y4Ri8= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Det udtales "4 fakultet" eller bare "4 udr\303\245bstegn". Man siger ogs\303\245, at antallet af m\303\245der, hvorp\303\245 man kan permutere 4 forskellige elementer, er LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JVEiNEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEiIUYnRi9GMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHRklGL0Yy, som er 24. I matematikken er der ogs\303\245 en formel til beregning af antal permutationer: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men der er alts\303\245 ingen grund til at bruge denne formel, n\303\245r man kan skrive LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiIUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdGTkYyRjw= direkte.NB! Skulle man f\303\245 lyst til at udskrive de 10 forskellige mulige r\303\246kkef\303\270lger, kan det g\303\270res via kommandoen permute fra pakken combinat :LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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Eksempel 1Givet et fodboldhold med 11 udtagne spillere. Hvor mange m\303\245der kan man rykke rundt p\303\245 spillernes positioner? (alle spilpositioner er forskellige). Svar :LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEjMTFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIhRidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZHLyUrZXhlY3V0YWJsZUdGOEYv = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Der er alts\303\245 39916800 forskellige m\303\245der.KombinationerVi har ofte brug for antallet af m\303\245der, hvorp\303\245 man kan udtr\303\246kke r elementer ud af en m\303\246ngde best\303\245ende af n forskellige elementer. R\303\246kkef\303\270lgen af udtr\303\246kningen er ligegyldig. Det kunne fx v\303\246re antallet af m\303\245der, hvorp\303\245 man kan udtr\303\246kke to elementer ud af m\303\246ngden 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. Svaret er, at det kan g\303\270res p\303\245LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpYmlub21pYWxGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYkLUYjNictSSNtbkdGJDYkUSI1RidGMi1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjs2JFEiMkYnRjIvJStleGVjdXRhYmxlR0YxRjJGMkZaRjI= = JSFHforskellige m\303\245der. I matematikb\303\270ger skrives det ofte 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 eller 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. Man kan dog ogs\303\245 skrive det s\303\245dan i Maple, hvis man henter symbolet 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 ovre i Expression paletten :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 = I matematikken er der ogs\303\245 en formel til beregning af antal udtr\303\246kninger: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men der er alts\303\245 ingen grund til at bruge denne formel, n\303\245r man kan bruge den mere direkte m\303\245de at g\303\270re det p\303\245 i Maple, n\303\246vnt ovenfor. NB! Skulle man f\303\245 lyst til at udskrive de 10 forskellige mulige kombinationer kan det g\303\270res via kommandoen choose fra pakken combinat :LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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Eksempel 2En fodboldtrup er p\303\245 25 spillere. P\303\245 hvor mange m\303\245der kan tr\303\246neren udtage de 11 spillere, som skal spille? (underforst\303\245et at spilpositionen ikke er v\303\246sentlig)Svar :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 = JSFHDer er alts\303\245 4457400 forskellige m\303\245der at udtage de 11 spillere p\303\245. PermutationerEn lidt anden situation end ovenn\303\246vnte med kombinationer er den, hvor det ikke er ligegyldigt i hvilken r\303\246kkef\303\270lge elementerne udtr\303\246kkes. Mens udtr\303\246kningen af f\303\270rst a og derefter b fra m\303\246ngden M ovenfor og udtr\303\246kningen af f\303\270rst b og dern\303\246st a fra M t\303\246ller som den samme udtr\303\246kning ovenfor, s\303\245 t\303\246ller det som to forskellige udtr\303\246kninger i tilf\303\246ldet med permutationer. R\303\246kkef\303\270lgen af udtr\303\246kningen er med andre ord ikke ligegyldig her. Det giver flere muligheder.I matematikken betegnes antallet med LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNigtRi82JlEibkYnRjJGNUY4LUkjbW9HRiQ2LlEiLEYnRjUvRjlRJ25vcm1hbEYnLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRi82JlEickYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y1RkQ= eller LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEibkYnRi9GMkY1LUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEickYnRi9GMkY1RjJGREYyRkRGMkZE og en formel for antallet er givet ved: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Sjovt nok har Maple ikke en indbygget kommando til det. Man kan dog selv definere den s\303\245ledes: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 = S\303\245 der er alts\303\245 20 mulige m\303\245der at udtr\303\246kke 2 elementer ud af 5 mulige, n\303\245r r\303\246kkef\303\270lgende betyder noget. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Eksempel 3En fodboldtrup er p\303\245 25 spillere. P\303\245 hvor mange m\303\245der kan tr\303\246neren udtage de 11 spillere, der skal spille, n\303\245r spilpositionerne tages i betragtning? (alle spilpositioner er forskellige). Svar :LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Der er alts\303\245 LUkjbW5HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JFEwMTc3OTI1MTQ0MzIwMDAwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn forskellige m\303\245der at udv\303\246lge det spillende hold med spilpositioner p\303\245. Binomialfordelingen via formelEksperiment : En basiseksperiment udf\303\270res n gange. Man er udelukkende interesseret i, hvor mange gange basiseksperimentet "lykkes". For det enkelte basiseksperiment er der kun to muligheder: Enten lykkes det eller ogs\303\245 mislykkes det. Sandsynligheden for at basiseksperimentet lykkes, betegnes p og kaldes for basissandsynligheden. Udfaldene af de enkelte basiseksperimenter skal v\303\246re uafh\303\246ngige. Stokastisk variabel X : Angiver hvor mange gange basiseksperimentet lykkes. Man kan vise, at sandsynligheden for at basiseksperimentet lykkes r gange er givet ved: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Man kan ogs\303\245 vise, at middelv\303\246rdien for den stokastiske variabel X er givet ved: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og at variansen er givet ved: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Eksempel 4Der sl\303\245s 17 gange med en terning. a) Hvad er sandsynligheden for at f\303\245 netop 2 femmere. b) Bestem sandsynligheden for at sl\303\245 h\303\270jst 4 femmere. c) Bestem sandsynligheden for at sl\303\245 mindst \303\251n femmer.d) Bestem sandsynligheden for at sl\303\245 mindst 2 og h\303\270jst 4 femmere. e) Hvor mange femmere vil man i snit f\303\245?f) Bestem variansen og spredningen for den stokastiske variabel X. L\303\270sning :Vi starter beh\303\246ndigt med at definere en funktion LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEickYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQEYyRkA=, som give sandsynligheden for at f\303\245 netop 2 femmere ved 17 kast med en terning.a) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUkjbW5HRiQ2JFEiMkYnL0YzUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRj5GPkZARj4= = Sandsynligheden for at f\303\245 netop 2 femmere ved kast med 17 terninger er alts\303\245 24.5%.b)Vi l\303\246gger sandsynligheden for 0, 1, 2, 3 og 4 femmere sammen: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 = eller ogs\303\245 via en sum: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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Sandsynligheden for at f\303\245 h\303\270jst 4 femmere ved kast med 17 terninger er alts\303\245 86.0%.c)Det modsatte af at f\303\245 mindst \303\251n femmer er at for ingen femmere. Derfor beregner vi sidstn\303\246vnte og tr\303\246kker resultatet fra 1. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKCZtaW51cztGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSJQRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JFEiMEYnRi8vJStleGVjdXRhYmxlR0Y4Ri9GL0ZlbkYv = Sandsynligheden for at f\303\245 mindst \303\251n femmer ved kast med 17 terninger er alts\303\245 95.4%.d)Vi l\303\246gger sandsynlighederne for 2, 3 og 4 femmere sammen: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 = Sandsynligheden for at sl\303\245 mindst 2 og h\303\270jst 4 femmere ved 17 slag med en terning er alts\303\245 66.2%. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=e) Vi bruger formlen for middelv\303\246rdien: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 = LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRLWF0fjEwfmRpZ2l0c0YnRi9GMkY1RkU=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Man vil alts\303\245 i snit f\303\245 2.83 femmere ved 17 kast med en terning. f)Vi bruger formlerne for variansen for spredningen: 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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Variansen er alts\303\245 2.36 og spredningen 1.54 ved 17 kast med en terning. Binomialfordelingen via Maple kommandoerDenne gang vil vi bruge de indbyggede kommandoer i Maple. F\303\270rst definerer man X til at v\303\246re enbinomialt stokastisk variabel med parametre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIzE3RidGMkY8RjJGPA== og Eksempel 5Der sl\303\245s 17 gange med en terning. a) Hvad er sandsynligheden for at f\303\245 netop 2 femmere. b) Bestem sandsynligheden for at sl\303\245 h\303\270jst 4 femmere. c) Bestem sandsynligheden for at sl\303\245 mindst \303\251n femmer.d) Bestem sandsynligheden for at sl\303\245 mindst 2 og h\303\270jst 4 femmere. e) Hvor mange femmere vil man i snit f\303\245?f) Bestem variansen og spredningen for den stokastiske variabel X. Ekstra:g) Tegn forskellige plots m.m. L\303\270sning :F\303\270rst defineres X til at v\303\246re en a)F\303\270rst angiver vi v\303\246rdierne af n og p. Dern\303\246st indf\303\270rer vi en stokastisk variabel (hedder random variable p\303\245 engelsk) X, som skal v\303\246re af typen binomialtfordelt stokastisk varaibel med parametre n og p. Herefter kan sandsynligheden for at f\303\245 netop k femmere ved kast med n terninger f\303\245s via kommandoen 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. Husk at kalde pakken Statistics f\303\270rst.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRK1N0YXRpc3RpY3NGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjpGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVkY9RkA=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Sandsynligheden for at f\303\245 netop 2 femmere ved kast med 17 terninger er alts\303\245 24.5%.b)Vi skal g\303\270re brug af fordelingsfunktionen, som angiver de kumulerede sandsynligheder (p\303\245 engelsk Cumulative Distribution Function). LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEkQ0RGRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRiw2JVEiWEYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictSSNtbkdGJDYkUSI0RidGQS8lK2V4ZWN1dGFibGVHRkVGQUZBRmZuRkE= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Sandsynligheden for at f\303\245 h\303\270jst 4 femmere ved kast med 17 terninger er alts\303\245 86.0%.c)Det modsatte af at f\303\245 mindst \303\251n femmer er at for ingen femmere. Derfor beregner vi sidstn\303\246vnte og tr\303\246kker resultatet fra 1. 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 = Sandsynligheden for at f\303\245 mindst \303\251n femmer ved kast med 17 terninger er alts\303\245 95.4%.d)Vi kan bruge den fordelingsfunktionen: F\303\270rst bestemmes sandsynligheden for at X er mindre end eller lig med 4. Men da har vi f\303\245et for meget med, s\303\245 vi tr\303\246kker sandsynligheden for at f\303\245 1 eller f\303\246rre femmere fra (bem\303\246rk, at det er 1, ikke 2 - overvej hvorfor). 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 = Sandsynligheden for at sl\303\245 mindst 2 og h\303\270jst 4 femmere ved 17 slag med en terning er alts\303\245 66.2%. JSFHe)Middelv\303\246rdien kaldes mean p\303\245 engelsk. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElTWVhbkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIlhGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQEY9RkA= = Man vil alts\303\245 i snit f\303\245 2.83 femmere ved 17 kast med en terning. f)Spredningen hedder standard deviation p\303\245 engelsk. P\303\245 dansk ogs\303\245 standardafvigelsen. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpVmFyaWFuY2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJYRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkBGPUZA = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEyU3RhbmRhcmREZXZpYXRpb25GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJYRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkBGPUZA = Variansen er alts\303\245 2.36 og spredningen 1.54 ved 17 kast med en terning. EkstraMan kan tegne et pindediagram over sandsynlighedsfordelingen for den stokastiske variabel X :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Vi kan ogs\303\245 tegne fordelingsfunktionen 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 :LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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JSFHMan kan anvende kommandoen Quantile til for given y-v\303\246rdi p\303\245 ovenst\303\245ende "sumkurve" at give den tilh\303\270rende x-v\303\246rdi. Helt pr\303\246cist er den p'te Quantile defineret som 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, hvor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YyLUYsNiRRIjFGJ0YvLyUrZXhlY3V0YWJsZUdGOEYv. 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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Stikpr\303\270ver fra forskellige fordelingerVi skal se, hvordan vi kan tage en stikpr\303\270ve fra forskellige fordelinger. F\303\270rst ser vi p\303\245 en diskret uniform fordeling af hele tal fra 1 til 6. Det kan t\303\246nkes p\303\245 som en ternings visning. Det uniforme g\303\245r p\303\245, at alle udfald i udfaldsrummet 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 er lige sandsynlige. De har alle sandsynligheden 1/6. For det f\303\270rste kalder vi pakken Statistics, for at det virker. S\303\245 definerer vi en stokastisk variabel X med den n\303\246vnte fordeling. I \303\270vrigt hedder en stokastisk variabel p\303\245 engelsk en Random Variable. I Maple kan man tage en stikpr\303\270ve ved kommandoen Sample. Som parametre giver man den givne stokastiske variable samt et tal for, hvor stor stikpr\303\270ven skal v\303\246re. Bem\303\246rk at kommandoen LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEqcmFuZG9taXplRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQEY9RkA= anvendes for at man f\303\245r en ny stikpr\303\270ve. Ellers vil en genberegning resultere i den samme stikpr\303\270ve! Man kan tegne stikpr\303\270vens resultat i et histogram. Vi kan ogs\303\245 f\303\245 angivet de enkelte frekvenser ved at anvende Gym-pakke kommandoen frekvens. JSFHEn diskret uniform fordelingLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRK1N0YXRpc3RpY3NGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjpGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVkY9RkA=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JSFHMan kan pr\303\270ve at \303\270ge v\303\246rdien af N til noget st\303\270rre og derefter genberegne. S\303\245 skulle man gerne se, at histogrammet f\303\245r ca. lige h\303\270je s\303\270jler, svarende til at der kommer omtrent lige mange 1'ere, 2'ere, ..., 6'ere. De store tals lov er i spil! JSFHBinomialfordelingenLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRK1N0YXRpc3RpY3NGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjpGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVkY9RkA=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEqcmFuZG9taXplRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjpGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVkY9RkA=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJEd5bUYnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZALUkjbW9HRiQ2LVEiOkYnRkAvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGPy8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZWRj1GQA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkobWZlbmNlZEdGJDYmLUYjNiYtRiw2JVEkc2VxRidGL0YyLUZQNiQtRiM2LC1GLDYlUSJTRidGL0YyLUZQNiYtRiM2JS1GLDYlUSJpRidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5RjkvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJ0Zcby1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1JI21uR0YkNiRRIjFGJ0Y5LUY2Ni1RIy4uRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORlxwLUYsNiVRIk5GJ0YvRjJGX29GOUY5Rl9vRjlGOUZhb0Zkby1GNjYtUSI6RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUZfb0Y5LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpZnJla3ZlbnNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSJMRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1JI21uR0YkNiRRIjVGJ0ZBLyUrZXhlY3V0YWJsZUdGRUZBRkFGZm5GQQ==JSFHOvenst\303\245ende frekvenser er alts\303\245 stikpr\303\270ve-frekvenserne. Man kan sammenligne med de teoriske frekvenser, som man kan f\303\245 ved at inds\303\246tte i sandsynlighedsfunktionen. F\303\270rst laver vi en liste M, som indeholder de mulige v\303\246rdier af X, dvs. 0, 1, 2, ... , n. Her er n lig med 17. NB! Bem\303\246rk at der kan v\303\246re "huller" i listen med stikpr\303\270ve-frekvenser. Det sker, hvis der slet ikke er et eneste element i stikpr\303\270ven, som antager dem mulige v\303\246rdi!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Igen kan man pr\303\270ve at \303\270ge st\303\270rrelsen af stikpr\303\270ven, dvs. \303\270ge v\303\246rdien af N. G\303\270r man det, skulle stikpr\303\270ve-frekvenserne gerne n\303\246rme sig til de teoretiske ...JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=