連續均勻分布continuous-uniform-distribution

目錄

• 引言
• R實踐
• 參考

引言

當從一個連續的區間內，比如[a, b]，隨機選擇隨機數的時候，給出的隨機數的數值就服從連續均勻分布。其概率函數，也可以比較容易理解，就是分段函數，區間外的部分概率是0，區間內的部分概率是均勻的值，即區間長度的倒數。

連續均勻分布是指從a到b的連續區間內隨機選擇隨機數的概率分布，其密度函數定義如下:

比如，當 a = 1, b = 3時，連續均勻分布的概率函數f(x)的圖像是：

R實踐

• R中提供了dunif函數， dunif gives the density，也即dunif函數計算，取出某個值的概率，如下：

dunif(x, min = 0, max = 1, log = FALSE)

# 從下可以看出，概率值在指定的區間內，是均勻的。在指定區間之外，概率值就是0.
> dunif(x = 0.5, min = 0, max = 1)
[1] 1
> dunif(x = 0.2, min = 0, max = 1)
[1] 1
> dunif(x = 0, min = 0, max = 3)
[1] 0.333
> dunif(x = -1, min = 0, max = 3)
[1] 0
> dunif(x = 4, min = 0, max = 3)
[1] 0

# 從下可以看出，均勻概率值的大小即為 1/(b-a)
> dunif(x = 0.2, min = 0, max = 2)
[1] 0.5
> dunif(x = 0.2, min = 0, max = 3)
[1] 0.333

所以可以利用dunif函數繪制上圖的概率函數曲線，如下：

> a <- 1
> b <- 3
> x <- seq(0, 5, 0.01)
> y <- dunif(x  = x, min = a, max = b)
> plot(x, y, main = "Density of uniform distribution",xlab = "x",ylab = "Density",col = "red",pch = 19, type = 'l')

• R 中提供了punif 函數，punif gives the distribution function ，如下：

punif(q, min = 0, max = 1, lower.tail = TRUE, log.p = FALSE)

> punif(q = 0.5, min = 0, max = 1, lower.tail = TRUE, log.p = FALSE)
[1] 0.5
> punif(q = 0.5, min = 0, max = 2, lower.tail = TRUE, log.p = FALSE)
[1] 0.25
> punif(q = 0.5, min = 0, max = 3, lower.tail = TRUE, log.p = FALSE)
[1] 0.167
> punif(q = 0.2, min = 0, max = 1, lower.tail = TRUE, log.p = FALSE)
[1] 0.2
> punif(q = 0.1, min = 0, max = 1, lower.tail = TRUE, log.p = FALSE)
[1] 0.1

• R 中提供了runif 函數，是生成隨機數的函數。如下：

runif(n, min = 0, max = 1)

> runif(n = 3, min = 0, max = 1)
[1] 0.240 0.812 0.876
> runif(n = 3, min = 0, max = 10)
[1] 9.30 6.60 6.89

參考

• [Continuous Uniform Distribution(http://www.r-tutor.com/elementary-statistics/probability-distributions/continuous-uniform-distribution)