1、概率密度函數
定義：對任一個隨機變量X，存在一個函數f(x)，滿足以上條件，那么就說，f(x)是X的概率密度函數：
意義說明：描述隨機變量在某一個確定取值點的可能性的函數，或者說是瞬時增幅的一個函數，用微分定義如下：
2、累積分布函數
定義：對任一隨機變量X，對於任意給定值a，所有小於值a出現的概率和，就是隨機變量X的分布函數，分布函數可以唯一決定一個隨機變量，定義公式：
性質：（1）有界性；（2）單調性；（3）右連續性。
累積分布函數由於英文為Cumulative Distribution Function，所以經常簡稱為CDF。
3、分位數函數
定義：分位數函數是累積分布函數的反函數，也就是說，給定概率值，計算出隨機變量的取值（左側分位數）。
常用的有四個分布的分位數：
標准正態分布，qnorm(p, mean=0, sd=1)
Student’s (t) , qt(p,df=N,ncp=0)
卡方分布：qchisq(p, df=N,ncp=0)
Fisher-Snedecor：qf(p, df1,df2,ncp=0)
特例：四分位數
定義：四分位數是統計學中分位數的一種，即把所有的數值從小到大朴烈並分為四等分，處於三個分割點的數就是四分位數。
選值原則：樣本總量N，分位數y（百分數），令
（1）L是整數，取第L和L+1的平均值
（2）L不是整數，取下一個最近的整數（1.2取2）
4、隨機數函數
定義，從一個給定函數的的取值中隨機挑出一個自變量，輸出的是因變量的值。
5、幾個常見的隨機變量的四種函數形式：
（1）The Normal Distribution
Usage：
dnorm(x, mean = 0, sd = 1, log = FALSE)
pnorm(q, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)
qnorm(p, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)
rnorm(n, mean = 0, sd = 1)
Arguments：
| x,q |
vector of quantiles. |
| p |
vector of probabilities. |
| n |
number of observations. If length(n) > 1, the length is taken to be the number required. |
| mean |
vector of means. |
| sd |
vector of standard deviations. |
| log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
| lower.tail |
logical; if TRUE (default), probabilities are P[X ≤ x] otherwise, P[X > x] |
（2）卡方分布
Usage：
dchisq(x, df, ncp=0, log = FALSE)
pchisq(q, df, ncp=0, lower.tail = TRUE, log.p = FALSE)
qchisq(p, df, ncp=0, lower.tail = TRUE, log.p = FALSE)
rchisq(n, df, ncp=0)
Arguments：
| x, q |
vector of quantiles. |
| p |
vector of probabilities. |
| n |
number of observations. If length(n) > 1, the length is taken to be the number required. |
| df |
degrees of freedom (non-negative, but can be non-integer). |
| ncp |
non-centrality parameter (non-negative). |
| log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
| lower.tail |
logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x]. |
（3）F分布
Usage：
df(x, df1, df2, ncp, log = FALSE)
pf(q, df1, df2, ncp, lower.tail = TRUE, log.p = FALSE)
qf(p, df1, df2, ncp, lower.tail = TRUE, log.p = FALSE)
rf(n, df1, df2, ncp)
Arguments：
| x, q |
vector of quantiles. |
| p |
vector of probabilities. |
| n |
number of observations. If length(n) > 1, the length is taken to be the number required. |
| df1, df2 |
degrees of freedom. Inf is allowed. |
| ncp |
non-centrality parameter. If omitted the central F is assumed. |
| log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
| lower.tail |
logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x]. |
（4）T分布
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：
| x, q |
vector of quantiles. |
| p |
vector of probabilities. |
| n |
number of observations. If length(n) > 1, the length is taken to be the number required. |
| df |
degrees of freedom (> 0, maybe non-integer). df = Inf is allowed. |
| ncp |
non-centrality parameter delta; currently except for rt(), only for abs(ncp) <= 37.62. If omitted, use the central t distribution. |
| log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
| lower.tail |
logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x]. |