在概率论和统计学中，t-分布（t-distribution）用于根据小样本来估计呈正态分布且方差未知的总体的均值。如果总体方差已知（例如在样本数量足够多时），则应该用正态分布来估计总体均值。

t分布曲线形态与n（确切地说与自由度df）大小有关。与标准正态分布曲线相比，自由度df越小，t分布曲线愈平坦，曲线中间愈低，曲线双侧尾部翘得愈高；自由度df愈大，t分布曲线愈接近正态分布曲线，当自由度df=∞时，t分布曲线为标准正态分布曲线。

The Student t Distribution

Description

Density, distribution function, quantile function and random generation for the t distribution with df degrees of freedom (and optional non-centrality parameter ncp).

Usage

dt(x, df, ncp, log = FALSE)
pt(q, df, ncp, lower.tail = TRUE, log.p = FALSE)
qt(p, df, ncp, lower.tail = TRUE, log.p = FALSE)
rt(n, df, ncp)

Arguments

####t分布
# 1.t分布中抽样函数rt
# location：x0; scale:gamma
n = 100
df <- 10
rt(n, df=df)

# 2.t分布概率密度函数
x <- seq(-10,10,0.1)
y <- dt(x,df)
plot(x,y)

# 3.t分布累积概率
# x <- seq(-20,20,0.1)
# plot(x,dt(x,df))
# P[X ≤ x]
pt(1,df=df)
# P[X > x]
pt(1,df=df,lower.tail = FALSE)

# probabilities p are given as log(p).
pt(1,df=df,log.p = TRUE)

# 4.qt函数(pt的反函数)
# 累积概率为0.95时的x值
# x <- seq(-10,10,0.1)
# plot(x,pt(x,df))
qt(0.95, df=df)
qt(0.995, df=df)

t分布的推导

那我们来写写代码，实践这个过程。我设定一个总体均数为0，标准差=1，样本量为1000的人群（图A是这个总体的概率分布）。图B，也是一次抽取三个人，抽了200个，图C一次抽取6个人。都分别计算t值。我们看到图C，一次抽取6个人，分布更多集中在总体均数为0。 还有一个，问题，图B中画绿框的这一部分，怎么解释呢？课后可以想想。（1.抽样误差、2.开端的，无闭合）

继续写代码实践，。如一次，，，他们分布会有什么特点。从图片可以得出什么规律：发现一次抽取3次和一次抽取6次比。一次抽取3次的曲线更低，向两边散。一次抽取6个样的，更集中。

#生成正态分布的数值,共1000,总体均数为0，标准差为1
data <- rnorm(1000, mean = 0, sd = 1)
#查看直方图分布
hist(data)
hist(data,freq = FALSE,,ylim=c(0,0.5))
lines(density(data),col="blue",lwd=2)

#如果抽取例数n=3的样本k个,假设抽取200个
data_n3_k200 = matrix(NA, nrow = 200, ncol = 3)
for (i in 1:200) {
  data_n3_k200[i, ] = sample(data, size = 3)
}
data_n3_k200 <- as.data.frame(data_n3_k200)
data_n3_k200$mean <- apply(data_n3_k200[1:200,1:3],1,mean)
data_n3_k200$sd <- apply(data_n3_k200[1:200,1:3],1,sd)
data_n3_k200$t <- (data_n3_k200$mean-0)/(data_n3_k200$sd/sqrt(2))
hist(data_n3_k200$t,freq = FALSE,ylim=c(0,0.6),xlim=c(-6,6))
lines(density(data_n3_k200$t),col="blue",lwd=2)


#如果抽取例数n=6的样本k个,假设抽取200个
data_n2_k200 = matrix(NA, nrow = 200, ncol = 6)
for (i in 1:200) {
  data_n2_k200[i, ] = sample(data, size = 6)
}
data_n2_k200 <- as.data.frame(data_n2_k200)
data_n2_k200$mean <- apply(data_n2_k200[1:200,1:6],1,mean)
data_n2_k200$sd <- apply(data_n2_k200[1:200,1:6],1,sd)
data_n2_k200$t <- (data_n2_k200$mean-0)/(data_n2_k200$sd/sqrt(2))
hist(data_n2_k200$t,freq = FALSE,ylim=c(0,0.9))
lines(density(data_n2_k200$t),col="blue",lwd=2)


#如果抽取例数n=3的样本k个,假设抽取10000次
data_n3_k10000 = matrix(NA, nrow = 10000, ncol = 3)
for (i in 1:10000) {
  data_n3_k10000[i, ] = sample(data, size = 3)
}
data_n3_k10000 <- as.data.frame(data_n3_k10000)
data_n3_k10000$mean <- apply(data_n3_k10000[1:10000,1:3],1,mean)
data_n3_k10000$sd <- apply(data_n3_k10000[1:10000,1:3],1,sd)
data_n3_k10000$t <- (data_n3_k10000$mean-0)/(data_n3_k10000$sd/sqrt(2))
hist(data_n3_k10000$t,freq = FALSE,ylim=c(0,0.5))
lines(density(data_n3_k10000$t),col="blue",lwd=2)zz


#如果抽取例数n=6的样本k个,假设抽取1000次
data_n2_k10000 = matrix(NA, nrow = 10000, ncol = 6)
for (i in 1:10000) {
  data_n2_k10000[i, ] = sample(data, size = 6)
}
data_n2_k10000 <- as.data.frame(data_n2_k10000)
data_n2_k10000$mean <- apply(data_n2_k10000[1:10000,1:6],1,mean)
data_n2_k10000$sd <- apply(data_n2_k10000[1:10000,1:6],1,sd)
data_n2_k10000$t <- (data_n2_k10000$mean-0)/(data_n2_k10000$sd/sqrt(2))
hist(data_n2_k10000$t,freq = FALSE,ylim=c(0,0.8))
lines(density(data_n2_k10000$t),col="blue",lwd=2)z
#hist(data_n3_k10000$t,freq = FALSE,ylim=c(0,0.5),xlim = c(-20,20))
plot(0,0.5,xlim=c(-1.5,1.5),ylim=c(0,0.7))
lines(density(data_n3_k10000$t),col="BLUE",lwd=2)
par(new=TRUE)
#hist(data_n2_k10000$t,freq = FALSE,axes = FALSE,xlab = "", ylab = "")
lines(density(data_n2_k10000$t),col="red",lwd=2)

R语言t分布函数Student t distribution（dt, pt, qt & rt ）实战

目录

R语言t分布函数Student t distribution（dt, pt, qt & rt ）实战