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November 17, 2015 11:47
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R Skript Stochastik Aufgabe 3 - Blatt 5
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#!/usr/bin/env Rscript | |
# Diese Funktion berechnet die exakte und die approximierte Version | |
calc_exact_and_approx <- function(n, p, x1, x2) { | |
# exakte Berechnung: | |
sum <- 0 | |
for(i in x1:x2) { | |
sum <- sum + ( choose(n, i) * p**i * (1 - p)**(n-i) ) | |
} | |
# Gibt die exakte Loesung aus | |
print(sprintf("Exakte Loesung: %f", sum)) | |
# Berechnung mit Normalapproximation mit Stetigkeitskorrektur: | |
# überprüfen ob die Approximation annährend gut ist: | |
# Am besten gilt: Varianz = np(1-p) >= 9. | |
# Aber es MUSS gelten: np >= 4. | |
approx_works = function(n, p) return(n * p * (1 - p) >= 9) | |
approx_must_work = function(n, p) return (n * p >= 4) | |
if (!approx_works(n, p)) { | |
stopifnot(approx_must_work(n, p)) | |
} | |
mi = n * p | |
sigma = sqrt( n * p * (1 - p) ) | |
sum_approx = pnorm((x2 + 0.5 - mi) / sigma) - pnorm((x1 - 0.5 - mi) / sigma) | |
print(sprintf("Approx Loesung: %f", sum_approx)) | |
print('') | |
} | |
# P(9 <= Y <= 11) fuer Y = Bin(20, 0.5) | |
calc_exact_and_approx(20, 0.5, 9, 11) | |
# "Exakte Loesung: 0.496555" | |
# "Approx Loesung: 0.497665" | |
# P(90 <= Y <= 110) fuer Y = Bin(200, 0.5) | |
calc_exact_and_approx(200, 0.5, 90, 110) | |
# "Exakte Loesung: 0.862633" | |
# "Approx Loesung: 0.862436" | |
# P(20 <= Y <= 40) fuer Y = Bin(200, 0.15) | |
calc_exact_and_approx(200, 0.15, 20, 40) | |
# "Exakte Loesung: 0.963120" | |
# "Approx Loesung: 0.962411" | |
# P(0 <= Y <= 20) fuer Y = Bin(200, 0.05) | |
calc_exact_and_approx(200, 0.05, 0, 20) | |
# "Exakte Loesung: 0.998840" | |
# "Approx Loesung: 0.999342" |
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