Hình:Normal approximation to binomial.png
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Plot of the probability density function of a normal distribution approximating the probability mass function of a binomial distribution
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Note: "Norm(12, 3)" is a normal distribution with a mean of 12 and a standard deviation of 3. Its variance is 48 × 1/4 × 3/4 = 9.
gnuplot source under GPL:
# normal (Gaussian) distribution
_ln_dnorm(x, m, s) = -0.5 * log(2*pi) - log(s) - 0.5*((x-m)*1.0/s)**2
dnorm(x, mean, sd) = exp(_ln_dnorm(x, mean, sd))
pnorm(x, mean, sd) = norm((x-mean) * 1.0/sd)
# binomial distribution
_ln_binom(x, n, p) =\
lgamma(n+1) - lgamma(x+1) - lgamma(n-x+1) + x*log(p) + (n-x)*log(1-p)
dbinom(x, size, prob) = (x==int(x))? exp(_ln_binom(floor(x), size, prob)) : 0
pbinom(x, size, prob) =\
(x<0)? 0 : (x<size)? ibeta(size-floor(x), floor(x)+1, 1-prob) : 1
set terminal postscript enhanced color solid lw 2 "Times-Roman" 20
set output
set key 22,0.13
n = 48
p = 0.25
xmax = 25
set samples 50*xmax+1
plot [0:xmax] \
dbinom(x, n, p) with impulses title "Binom(48, 0.25)", \
dnorm(x, n*p, sqrt(n*p*(1-p))) linetype 3 title "Norm(12, 3)"
| This work is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This work is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. |
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