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●平均0, 標準偏差0.8の正規分布の確率密度関数（dnorm）
> x <- seq(-3, 3, 0.05)
> plot(x,dnorm(x, mean=0, sd=0.4), type="n")
> curve(dnorm(x, mean=0, sd=0.8), type="l",add=T)
●正規分布の確率分布関数（pnorm）とその逆関数（qnorm）
> curve(pnorm(x, mean=0, sd=0.8), type="l", lty=3, add=T)
●５％点の表示
> abline(h=0.05)
> lower.alpha5 <- qnorm(0.05, mean=0, sd=0.8)
> lower.alpha5
[1] -1.315883
> abline(v=lower.alpha5)
> points(lower.alpha5, 0.05, cex=3.0, pch="*")
> abline(h=0.95)
> upper.alpha5 <- qnorm(0.05, mean=0, sd=0.8, lower.tail = FALSE)
> upper.alpha5
[1] 1.315883
> abline(v=upper.alpha5)
> points(upper.alpha5, 0.95, cex=3.0, pch="*")
●１％点の表示
> abline(h=0.01, lty=2)
> lower.alpha1 <- qnorm(0.01, mean=0, sd=0.8)
> lower.alpha1
[1] -1.861078
> abline(v=lower.alpha1, lty=2)
> points(lower.alpha1, 0.01, cex=3.0, pch="*")
> abline(h=0.99, lty=2)
> upper.alpha1 <- qnorm(0.01, mean=0, sd=0.8, lower.tail = FALSE)
> upper.alpha1
[1] 1.861078
> abline(v=upper.alpha1, lty=2)
> points(upper.alpha1, 0.99, cex=3.0, pch="*")
●正規乱数（rnorm）の生成とヒストグラム表示
> random.norm <- rnorm(10, mean=0, sd=0.8)
> hist(random.norm, freq=F)
> random.norm <- rnorm(1000, mean=0, sd=0.8)
> hist(random.norm, freq=F)
> random.norm <- rnorm(1000000, mean=0, sd=0.8)
> hist(random.norm, freq=F)
> curve(dnorm(x, mean=0, sd=0.8), add=T)
●正規分布のパラメーター（１）――平均μを変える
> x <- seq(-4, 4, 0.01)
> plot(x, dnorm(x, mean=0, sd=0.5), type="n")
> title("Normal Distribution\nmean=0 -> 2.0")
> for (i in 1:5) curve(dnorm(x, mean=0.5*(i-1), sd=0.5), type="l", add=T)
●正規分布のパラメーター（２）――分散σ^2を変える
> x <- seq(-4, 4, 0.01)
> plot(x, dnorm(x, mean=0, sd=0.5), type="n")
> title("Normal Distribution\nsd=0.5 -> 2.5")
> for (i in 1:5) curve(dnorm(x, mean=0, sd=0.5*i), type="l", add=T)
●標準正規分布（平均０，分散１）
> mean1 <- 1.0
> sd2 <- 2.0
> plot(x, dnorm(x, mean=0, sd=1), type="n")
> x <- rnorm(10000, mean=mean1, sd=sd2)
> hist(x, freq=F, density=25, angle=135, add=T)
> curve(dnorm(x, mean=mean1, sd=sd2), type="l", lty=2, lwd=2, add=T)
> hist((x - mean1)/sd2, freq=F, density=25, angle=45, add=T)
> curve(dnorm(x, mean=0, sd=1), type="l", lty=1, lwd=2, add=T)
正規分布に関連する関数（dnorm, pnorm, qnorm, rnorm）
三中信宏
Copyright (c) 2004 by MINAKA Nobuhiro. All rights reserved.
●平均0, 標準偏差0.8の正規分布の確率密度関数（dnorm）
> x <- seq(-3, 3, 0.05)
> plot(x,dnorm(x, mean=0, sd=0.4), type="n")
> curve(dnorm(x, mean=0, sd=0.8), type="l",add=T)
●正規分布の確率分布関数（pnorm）とその逆関数（qnorm）
> curve(pnorm(x, mean=0, sd=0.8), type="l", lty=3, add=T)
●５％点の表示
> abline(h=0.05)
> lower.alpha5 <- qnorm(0.05, mean=0, sd=0.8)
> lower.alpha5
[1] -1.315883
> abline(v=lower.alpha5)
> points(lower.alpha5, 0.05, cex=3.0, pch="*")
> abline(h=0.95)
> upper.alpha5 <- qnorm(0.05, mean=0, sd=0.8, lower.tail = FALSE)
> upper.alpha5
[1] 1.315883
> abline(v=upper.alpha5)
> points(upper.alpha5, 0.95, cex=3.0, pch="*")
●１％点の表示
> abline(h=0.01, lty=2)
> lower.alpha1 <- qnorm(0.01, mean=0, sd=0.8)
> lower.alpha1
[1] -1.861078
> abline(v=lower.alpha1, lty=2)
> points(lower.alpha1, 0.01, cex=3.0, pch="*")
> abline(h=0.99, lty=2)
> upper.alpha1 <- qnorm(0.01, mean=0, sd=0.8, lower.tail = FALSE)
> upper.alpha1
[1] 1.861078
> abline(v=upper.alpha1, lty=2)
> points(upper.alpha1, 0.99, cex=3.0, pch="*")
●正規乱数（rnorm）の生成とヒストグラム表示
> random.norm <- rnorm(10, mean=0, sd=0.8)
> hist(random.norm, freq=F)
> random.norm <- rnorm(1000, mean=0, sd=0.8)
> hist(random.norm, freq=F)
> random.norm <- rnorm(1000000, mean=0, sd=0.8)
> hist(random.norm, freq=F)
> curve(dnorm(x, mean=0, sd=0.8), add=T)
●正規分布のパラメーター（１）――平均μを変える
> x <- seq(-4, 4, 0.01)
> plot(x, dnorm(x, mean=0, sd=0.5), type="n")
> title("Normal Distribution\nmean=0 -> 2.0")
> for (i in 1:5) curve(dnorm(x, mean=0.5*(i-1), sd=0.5), type="l", add=T)
●正規分布のパラメーター（２）――分散σ^2を変える
> x <- seq(-4, 4, 0.01)
> plot(x, dnorm(x, mean=0, sd=0.5), type="n")
> title("Normal Distribution\nsd=0.5 -> 2.5")
> for (i in 1:5) curve(dnorm(x, mean=0, sd=0.5*i), type="l", add=T)
●標準正規分布（平均０，分散１）
> mean1 <- 1.0
> sd2 <- 2.0
> plot(x, dnorm(x, mean=0, sd=1), type="n")
> x <- rnorm(10000, mean=mean1, sd=sd2)
> hist(x, freq=F, density=25, angle=135, add=T)
> curve(dnorm(x, mean=mean1, sd=sd2), type="l", lty=2, lwd=2, add=T)
> hist((x - mean1)/sd2, freq=F, density=25, angle=45, add=T)
> curve(dnorm(x, mean=0, sd=1), type="l", lty=1, lwd=2, add=T)
Last Modified: 8 December 2004 by MINAKA Nobuhiro