中心極限定理

作成日

2023年8月29日

library(tidyverse)
set.seed(19861008)

正規分布の例

  • \(\mu\)と\(\sigma\)
  • 期待値は\(\mu\)
  • 分散は\(\sigma^2\)であり、標準偏差は\(\sigma\)
  • \(\mu = 10\)、\(\sigma = 5\)の正規分布（ 図 1 ）の場合、期待値は10、標準偏差は5
Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
ℹ Please use `linewidth` instead.

図 1: 正規分布の確率密度曲線（\(\mu\) = 10、\(\sigma\) = 5）
vec1 <- rep(NA, 10000)
for (i in 1:10000) {
    vec1[i] <- mean(rnorm(1, mean = 10, sd = 5))
}

vec1 %>%
    enframe(name = "trial", value = "mean") %>%
    ggplot() +
    geom_histogram(aes(x = mean), color = "white") +
    geom_vline(xintercept = 10, color = "red") +
    coord_cartesian(xlim = c(-5, 25)) +
    labs(x = "標本平均", y = "度数") +
    theme_bw()

mean(vec1)
[1] 9.993612
sd(vec1)
[1] 5.052632
vec2 <- rep(NA, 10000)
for (i in 1:10000) {
    vec2[i] <- mean(rnorm(10, mean = 10, sd = 5))
}

vec2 %>%
    enframe(name = "trial", value = "mean") %>%
    ggplot() +
    geom_histogram(aes(x = mean), color = "white") +
    geom_vline(xintercept = 10, color = "red") +
    coord_cartesian(xlim = c(-5, 25)) +
    labs(x = "標本平均", y = "度数") +
    theme_bw()

mean(vec2)
[1] 9.994122
sd(vec2)
[1] 1.584378
vec3 <- rep(NA, 10000)
for (i in 1:10000) {
    vec3[i] <- mean(rnorm(100, mean = 10, sd = 5))
}

vec3 %>%
    enframe(name = "trial", value = "mean") %>%
    ggplot() +
    geom_histogram(aes(x = mean), color = "white") +
    geom_vline(xintercept = 10, color = "red") +
    coord_cartesian(xlim = c(-5, 25)) +
    labs(x = "標本平均", y = "度数") +
    theme_bw()

mean(vec3)
[1] 10.00129
sd(vec3)
[1] 0.4980742

一様分布の例

  • 最小値（\(a\)）と最大値（\(b\)）
  • 期待値は\(\frac{a + b}{2}\)
  • 分散は\(\frac{(b - a)^2}{12}\)であり、標準偏差は\(\frac{b - a}{\sqrt{12}}\)
  • \(a = 5\)、\(b = 15\)の一様分布（ 図 2 ）の場合、期待値は10、標準偏差は約2.89

図 2: 一様分布の確率密度曲線（\(a\) = 5、\(b\) = 15）
vec4 <- rep(NA, 10000)
for (i in 1:10000) {
    vec4[i] <- mean(runif(100, min = 5, max = 15))
}

vec4 %>%
    enframe(name = "trial", value = "mean") %>%
    ggplot() +
    geom_histogram(aes(x = mean), color = "white") +
    geom_vline(xintercept = 10, color = "red") +
    labs(x = "標本平均", y = "度数") +
    theme_bw()

mean(vec4)
[1] 9.999566
sd(vec4)
[1] 0.2892989

ポアソン分布の例

  • \(\lambda\)のみ
  • 期待値は\(\lambda\)
  • 分散は\(\lambda\)であり、標準偏差は\(\sqrt{\lambda}\)
  • \(\lambda = 10\)のポアソン分布（ 図 3 ）の場合、期待値は10、標準偏差は約3.16

図 3: ポアソン分布の確率密度曲線（\(\lambda\) = 10）
vec5 <- rep(NA, 10000)
for (i in 1:10000) {
    vec5[i] <- mean(rpois(100, lambda = 10))
}

vec5 %>%
    enframe(name = "trial", value = "mean") %>%
    ggplot() +
    geom_histogram(aes(x = mean), color = "white") +
    geom_vline(xintercept = 10, color = "red") +
    labs(x = "標本平均", y = "度数") +
    theme_bw()

mean(vec5)
[1] 10.00047
sd(vec5)
[1] 0.3174378

ガンマ分布の例

  • 形状パラメーター（\(k\)）と尺度パラメーター（\(\theta\)）
  • 期待値は\(k\theta\)
  • 分散は\(k\theta^2\)であり、標準偏差は\(\sqrt{k}\theta\)
  • \(k = 2\)、\(\theta = 5\)のガンマ分布（ 図 4 ）の場合、期待値は10、標準偏差は約7.07

図 4: ガンマ分布の確率密度曲線（\(k\) = 2、\(\theta\) = 5）
vec6 <- rep(NA, 10000)
for (i in 1:10000) {
    vec6[i] <- mean(rgamma(100, shape = 2, scale = 5))
}

vec6 %>%
    enframe(name = "trial", value = "mean") %>%
    ggplot() +
    geom_histogram(aes(x = mean), color = "white") +
    geom_vline(xintercept = 10, color = "red") +
    labs(x = "標本平均", y = "度数") +
    theme_bw()

mean(vec6)
[1] 10.00008
sd(vec6)
[1] 0.7052151