R version 4.2.3 (2023-03-15 ucrt) -- "Shortstop Beagle"
Copyright (C) 2023 The R Foundation for Statistical Computing
Platform: x86_64-w64-mingw32/x64 (64-bit)

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> e <- rexp(30)
> x <- rnorm(30)
> hist(x)
> hist(x,breaks=8)
> y <- rnorm(100)
> hist(y)
> hist(e)
> hist(e,probability=TRUE)
> t <- seq(0,5,0.2)
> lines(t,dexp(t))
> hist(x,probability=TRUE)
> lines(t,dexp(t))
> q <- seq(0,1,0.1)
> qe <- quantile(e,q)
> qe.elmeleti <- qexp(q)
> qe
         0%         10%         20%         30% 
0.006366025 0.032688550 0.210598279 0.327172176 
        40%         50%         60%         70% 
0.415412967 0.490363730 0.609699558 0.914633173 
        80%         90%        100% 
1.268915147 1.839480758 3.338329546 
> qe.elmeleti
 [1] 0.0000000 0.1053605 0.2231436 0.3566749
 [5] 0.5108256 0.6931472 0.9162907 1.2039728
 [9] 1.6094379 2.3025851       Inf
> qe - qe.elmeleti
          0%          10%          20% 
 0.006366025 -0.072671966 -0.012545272 
         30%          40%          50% 
-0.029502768 -0.095412657 -0.202783450 
         60%          70%          80% 
-0.306591173 -0.289339631 -0.340522765 
         90%         100% 
-0.463104335         -Inf 
> plot(qe,qe.elmeleti)
> abline(0,1)
> e2 <- rexp(100)
> qe2 <- quantile(e2,q)
> plot(qe2,qe.elmeleti)
> abline(0,1)
> qn.elmeleti <- qnorm(q)
> plot(qe2,qn.elmeleti)
> abline(0,1)
> y <- rnorm(100) * 2 + 3
> mean(y)
[1] 2.665655
> var(y)
[1] 3.731034
> qy <- quantile(y,q)
> plot(qy,qn.elmeleti)
> abline(0,1)
> abline(3,2,col="blue")
> plot(qn.elmeleti,qy)
> abline(0,1)
> abline(3,2,col="blue")
> ?qqplot
starting httpd help server ... done
> ?qqplot
> qqplot(e2,qexp(q))
> qqplot(e2,qexp)
Error in sort.int(x, na.last = na.last, decreasing = decreasing, ...) : 
  'x' must be atomic
> qqplot(e2,qexp(seq(0,1,0.05)))
> qqnorm(y)
> abline(3,2,col="blue")
> qqline(y,col="red")
> qqline(y,distribution=exp)
> qqplot(e2,qexp(seq(0,1,0.05)))
> qqline(y,distribution=exp)
> qqplot(e2,qexp(seq(0,1,0.05)))
> qqline(e2,distribution=exp)
> qqplot(e2,qexp(seq(0,1,0.05)))
> qqline(e2,distribution=qexp)
> x <- rnorm(30)
> y <- rnorm(30) + 3
> mean(x)
[1] -0.08588092
> mean(y)
[1] 3.095892
> x <- rnorm(30)
> y <- rnorm(30) + 3
> mean(x)
[1] 0.03593734
> mean(y)
[1] 3.055083
> x <- rnorm(30)
> y <- rnorm(30) + 3
> mean(x)
[1] -0.1406691
> mean(y)
[1] 3.139988
> x <- rnorm(30)
> y <- rnorm(30) + 3
> mean(x)
[1] -0.3526888
> mean(y)
[1] 3.123097
> plot(z,dt(z,df=1),type="l")
Error in plot(z, dt(z, df = 1), type = "l") : object 'z' not found
> z <- seq(-5,5,0.1)
> plot(z,dt(z,df=1),type="l")
> plot(z,dt(z,df=1),type="l")
> lines(z,dt(z,df=5),col="blue")
> lines(z,dt(z,df=20),col="red")
> lines(z,dt(z,df=Inf),col="green")
> plot(z,dt(z,df=1),type="l",ylim=c(0,1))
> lines(z,dt(z,df=5),col="blue")
> lines(z,dt(z,df=20),col="red")
> lines(z,dt(z,df=Inf),col="green")
> plot(z,dt(z,df=1),type="l",ylim=c(0,0.5))
> lines(z,dt(z,df=5),col="blue")
> lines(z,dt(z,df=20),col="red")
> lines(z,dt(z,df=Inf),col="green")
> plot(z,dt(z,df=29),type="l")
> z <- seq(-4,4,0.1)
> plot(z,dt(z,df=29),type="l")
> qt(c(0.05,0.95),df=29)
[1] -1.699127  1.699127
> pt(qt(c(0.05,0.95),df=29),df=29)
[1] 0.05 0.95
> abline(v=qt(c(0.05,0.95),df=29))
> tt <- t.test(x)
> tt

        One Sample t-test

data:  x
t = -1.9049, df = 29, p-value = 0.06675
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 -0.73135786  0.02598031
sample estimates:
 mean of x 
-0.3526888 

> tt$stat
        t 
-1.904907 
> abline(v=tt$stat,col="red")
> for (i in 1:20) {
+ x <- rnorm(30) ; 
+ abline(v=t.test(x)$stat,col="blue")
+ }
> plot(z,dt(z,df=29),type="l")
> qt(c(0.05,0.95),df=29)
[1] -1.699127  1.699127
> abline(v=qt(c(0.05,0.95),df=29))
> for (i in 1:20) {
+ x <- rnorm(30) ; 
+ abline(v=t.test(x)$stat,col="blue",lty=2)
+ }
> abline(v=t.test(y)$stat,col="purple")
> t.test(y)$stat
      t 
14.6773 
> tt

        One Sample t-test

data:  x
t = -1.9049, df = 29, p-value = 0.06675
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 -0.73135786  0.02598031
sample estimates:
 mean of x 
-0.3526888 

> abline(v=tt$stat,col="red")
> tt

        One Sample t-test

data:  x
t = -1.9049, df = 29, p-value = 0.06675
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 -0.73135786  0.02598031
sample estimates:
 mean of x 
-0.3526888 

> tt$stat
        t 
-1.904907 
> pt(tt$stat)
Error in pt(tt$stat) : argument "df" is missing, with no default
> pt(tt$stat,df=29)
         t 
0.03337634 
> pt(tt$stat,df=29)
         t 
0.03337634 
> 2*pt(tt$stat,df=29)
         t 
0.06675268 
> tt$p.value
[1] 0.06675268
> y <- rnorm(30) + 3
> t.test(y,mu=3)

        One Sample t-test

data:  y
t = 0.071018, df = 29, p-value = 0.9439
alternative hypothesis: true mean is not equal to 3
95 percent confidence interval:
 2.627463 3.399339
sample estimates:
mean of x 
 3.013401 

> t.test(y,mu=2.8)

        One Sample t-test

data:  y
t = 1.1309, df = 29, p-value = 0.2674
alternative hypothesis: true mean is not equal to 2.8
95 percent confidence interval:
 2.627463 3.399339
sample estimates:
mean of x 
 3.013401 

> ?t.test
> x <- rnorm(20) + 3
> t.test(x,y)

        Welch Two Sample t-test

data:  x and y
t = -0.24902, df = 46.789, p-value = 0.8044
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.5885560  0.4589112
sample estimates:
mean of x mean of y 
 2.948579  3.013401 

> z <- y - 1 + rnorm(30)
> t.test(y,z,paired=TRUE)

        Paired t-test

data:  y and z
t = 5.7248, df = 29, p-value = 3.397e-06
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
 0.8067382 1.7035538
sample estimates:
mean difference 
       1.255146 

> plot(t,dt(t,df=29),type="l")
> abline(v=qt(0.1,df=29))
> t <- seq(-3,3,0.2)
> plot(t,dt(t,df=29),type="l")
> abline(v=qt(0.1,df=29))
> t.test(z-y,alternative="less")

        One Sample t-test

data:  z - y
t = -5.7248, df = 29, p-value = 1.698e-06
alternative hypothesis: true mean is less than 0
95 percent confidence interval:
       -Inf -0.8826197
sample estimates:
mean of x 
-1.255146 

> t.test(y,z,alternative="less",paired=TRUE)

        Paired t-test

data:  y and z
t = 5.7248, df = 29, p-value = 1
alternative hypothesis: true mean difference is less than 0
95 percent confidence interval:
     -Inf 1.627672
sample estimates:
mean difference 
       1.255146 

> t.test(z,y,alternative="less",paired=TRUE)

        Paired t-test

data:  z and y
t = -5.7248, df = 29, p-value = 1.698e-06
alternative hypothesis: true mean difference is less than 0
95 percent confidence interval:
       -Inf -0.8826197
sample estimates:
mean difference 
      -1.255146 

> x <- round(runif(50))
> y <- round(runif(50))
> adat <- data.frame(x=x,y=y)
> adat
   x y
1  1 0
2  0 1
3  1 0
4  0 1
5  0 1
6  0 0
7  0 0
8  0 1
9  1 0
10 0 1
11 1 0
12 0 0
13 1 0
14 0 0
15 0 1
16 1 0
17 0 0
18 0 1
19 1 1
20 0 1
21 0 0
22 0 1
23 0 0
24 0 1
25 0 0
26 1 1
27 1 1
28 0 1
29 1 1
30 1 1
31 1 0
32 1 1
33 0 0
34 1 0
35 0 0
36 1 0
37 1 1
38 0 1
39 1 0
40 0 1
41 1 1
42 1 0
43 0 1
44 1 0
45 1 1
46 1 0
47 1 0
48 1 1
49 0 0
50 1 1
> adat$z <- x+y
> adat
   x y z
1  1 0 1
2  0 1 1
3  1 0 1
4  0 1 1
5  0 1 1
6  0 0 0
7  0 0 0
8  0 1 1
9  1 0 1
10 0 1 1
11 1 0 1
12 0 0 0
13 1 0 1
14 0 0 0
15 0 1 1
16 1 0 1
17 0 0 0
18 0 1 1
19 1 1 2
20 0 1 1
21 0 0 0
22 0 1 1
23 0 0 0
24 0 1 1
25 0 0 0
26 1 1 2
27 1 1 2
28 0 1 1
29 1 1 2
30 1 1 2
31 1 0 1
32 1 1 2
33 0 0 0
34 1 0 1
35 0 0 0
36 1 0 1
37 1 1 2
38 0 1 1
39 1 0 1
40 0 1 1
41 1 1 2
42 1 0 1
43 0 1 1
44 1 0 1
45 1 1 2
46 1 0 1
47 1 0 1
48 1 1 2
49 0 0 0
50 1 1 2
> table(x,y)
   y
x    0  1
  0 11 14
  1 14 11
> table(x)
x
 0  1 
25 25 
> table(y)
y
 0  1 
25 25 
> table(x,z)
Error in table(x, z) : all arguments must have the same length
> z <- adat$z
> table(x,z)
   z
x    0  1  2
  0 11 14  0
  1  0 14 11
> chisq.test(x,z)

        Pearson's Chi-squared test

data:  x and z
X-squared = 22, df = 2, p-value = 1.67e-05

> chisq.test(x,y)

        Pearson's Chi-squared test with Yates' continuity
        correction

data:  x and y
X-squared = 0.32, df = 1, p-value = 0.5716

> chisq.test(table(x,y))

        Pearson's Chi-squared test with Yates' continuity
        correction

data:  table(x, y)
X-squared = 0.32, df = 1, p-value = 0.5716

> a <- rnorm(100)
> b <- rnorm(100)
> c <- a+b
> ?cut
> a.int <- cut(a,breaks=2)
> a.int
  [1] (0.03,2.5]   (0.03,2.5]   (-2.44,0.03] (-2.44,0.03]
  [5] (0.03,2.5]   (-2.44,0.03] (-2.44,0.03] (-2.44,0.03]
  [9] (-2.44,0.03] (-2.44,0.03] (-2.44,0.03] (-2.44,0.03]
 [13] (0.03,2.5]   (0.03,2.5]   (-2.44,0.03] (-2.44,0.03]
 [17] (0.03,2.5]   (0.03,2.5]   (-2.44,0.03] (-2.44,0.03]
 [21] (0.03,2.5]   (0.03,2.5]   (-2.44,0.03] (-2.44,0.03]
 [25] (-2.44,0.03] (-2.44,0.03] (-2.44,0.03] (-2.44,0.03]
 [29] (0.03,2.5]   (0.03,2.5]   (0.03,2.5]   (-2.44,0.03]
 [33] (0.03,2.5]   (-2.44,0.03] (0.03,2.5]   (0.03,2.5]  
 [37] (0.03,2.5]   (-2.44,0.03] (-2.44,0.03] (-2.44,0.03]
 [41] (-2.44,0.03] (0.03,2.5]   (0.03,2.5]   (-2.44,0.03]
 [45] (-2.44,0.03] (-2.44,0.03] (0.03,2.5]   (0.03,2.5]  
 [49] (-2.44,0.03] (0.03,2.5]   (-2.44,0.03] (-2.44,0.03]
 [53] (0.03,2.5]   (0.03,2.5]   (-2.44,0.03] (0.03,2.5]  
 [57] (-2.44,0.03] (0.03,2.5]   (0.03,2.5]   (-2.44,0.03]
 [61] (-2.44,0.03] (-2.44,0.03] (0.03,2.5]   (-2.44,0.03]
 [65] (0.03,2.5]   (0.03,2.5]   (0.03,2.5]   (-2.44,0.03]
 [69] (0.03,2.5]   (0.03,2.5]   (0.03,2.5]   (-2.44,0.03]
 [73] (0.03,2.5]   (0.03,2.5]   (0.03,2.5]   (-2.44,0.03]
 [77] (-2.44,0.03] (-2.44,0.03] (0.03,2.5]   (-2.44,0.03]
 [81] (-2.44,0.03] (0.03,2.5]   (0.03,2.5]   (-2.44,0.03]
 [85] (-2.44,0.03] (-2.44,0.03] (0.03,2.5]   (0.03,2.5]  
 [89] (-2.44,0.03] (0.03,2.5]   (-2.44,0.03] (-2.44,0.03]
 [93] (-2.44,0.03] (0.03,2.5]   (0.03,2.5]   (0.03,2.5]  
 [97] (-2.44,0.03] (0.03,2.5]   (0.03,2.5]   (-2.44,0.03]
Levels: (-2.44,0.03] (0.03,2.5]
> table(a.int)
a.int
(-2.44,0.03]   (0.03,2.5] 
          53           47 
> b.int <- cut(b,breaks=seq(-2,2,1))
> b.int
  [1] (0,1]   (-2,-1] (-1,0]  (-2,-1] (0,1]   (-1,0]  (0,1]  
  [8] (-1,0]  (-1,0]  <NA>    (0,1]   (-1,0]  (-2,-1] (-2,-1]
 [15] (-1,0]  (-2,-1] <NA>    (-1,0]  (-1,0]  (1,2]   (-2,-1]
 [22] (0,1]   (1,2]   (-1,0]  (1,2]   (0,1]   (-2,-1] (1,2]  
 [29] (-1,0]  (-1,0]  (-1,0]  (1,2]   (-1,0]  (0,1]   (1,2]  
 [36] (1,2]   (1,2]   (1,2]   (-2,-1] (0,1]   (-2,-1] (0,1]  
 [43] (0,1]   (1,2]   (-2,-1] (-2,-1] (1,2]   (0,1]   (1,2]  
 [50] (0,1]   (1,2]   (1,2]   (0,1]   (0,1]   (1,2]   (-2,-1]
 [57] (0,1]   (-1,0]  (1,2]   (-2,-1] (1,2]   (-2,-1] (0,1]  
 [64] (0,1]   (0,1]   (-1,0]  (-2,-1] (0,1]   (-1,0]  (-2,-1]
 [71] (-2,-1] (0,1]   (0,1]   (0,1]   (0,1]   (0,1]   (-2,-1]
 [78] (-2,-1] (0,1]   (-1,0]  (-1,0]  (0,1]   (-2,-1] (1,2]  
 [85] (-2,-1] (0,1]   (0,1]   (-1,0]  (-1,0]  (1,2]   (-1,0] 
 [92] (-2,-1] (-1,0]  (-1,0]  (-1,0]  (-2,-1] (-1,0]  (-1,0] 
 [99] (-1,0]  (-1,0] 
Levels: (-2,-1] (-1,0] (0,1] (1,2]
> table(b.int)
b.int
(-2,-1]  (-1,0]   (0,1]   (1,2] 
     23      28      28      19 
> sum(table(b.int))
[1] 98
> range(c)
[1] -3.407855  4.404248
> min(c) ; max(c)
[1] -3.407855
[1] 4.404248
> seq(-4,5,length=3)
[1] -4.0  0.5  5.0
> seq(-4,5,length=4)
[1] -4 -1  2  5
> c.int <- cut(c,seq(-4,5,length=4))
> c.int
  [1] (2,5]   (-4,-1] (-1,2]  (-4,-1] (2,5]   (-4,-1] (-1,2] 
  [8] (-1,2]  (-4,-1] (-4,-1] (-1,2]  (-4,-1] (-1,2]  (-1,2] 
 [15] (-4,-1] (-4,-1] (-1,2]  (-1,2]  (-4,-1] (-1,2]  (-1,2] 
 [22] (-1,2]  (-1,2]  (-1,2]  (-1,2]  (-1,2]  (-4,-1] (-1,2] 
 [29] (-1,2]  (-1,2]  (-1,2]  (-1,2]  (-1,2]  (-1,2]  (2,5]  
 [36] (-1,2]  (-1,2]  (-1,2]  (-4,-1] (-1,2]  (-4,-1] (-1,2] 
 [43] (-1,2]  (-1,2]  (-4,-1] (-4,-1] (2,5]   (-1,2]  (-1,2] 
 [50] (-1,2]  (-1,2]  (-1,2]  (-1,2]  (-1,2]  (-1,2]  (-1,2] 
 [57] (-1,2]  (-1,2]  (-1,2]  (-4,-1] (-1,2]  (-4,-1] (-1,2] 
 [64] (-1,2]  (-1,2]  (-1,2]  (-1,2]  (-1,2]  (-1,2]  (-1,2] 
 [71] (-1,2]  (-1,2]  (-1,2]  (-1,2]  (-1,2]  (-1,2]  (-4,-1]
 [78] (-4,-1] (-1,2]  (-4,-1] (-1,2]  (-1,2]  (-1,2]  (-1,2] 
 [85] (-4,-1] (-1,2]  (-1,2]  (-1,2]  (-1,2]  (-1,2]  (-4,-1]
 [92] (-4,-1] (-1,2]  (-1,2]  (-1,2]  (-1,2]  (-4,-1] (-1,2] 
 [99] (-1,2]  (-4,-1]
Levels: (-4,-1] (-1,2] (2,5]
> chisq.test(a.int,b.int)

        Pearson's Chi-squared test

data:  a.int and b.int
X-squared = 4.0466, df = 3, p-value = 0.2565

> chisq.test(a.int,c.int)

        Pearson's Chi-squared test

data:  a.int and c.int
X-squared = 25.9, df = 2, p-value = 2.376e-06

Warning message:
In chisq.test(a.int, c.int) : Chi-squared approximation may be incorrect
> table(a.int,c.int)
              c.int
a.int          (-4,-1] (-1,2] (2,5]
  (-2.44,0.03]      23     30     0
  (0.03,2.5]         1     42     4
> table(a.int,b.int)
              b.int
a.int          (-2,-1] (-1,0] (0,1] (1,2]
  (-2.44,0.03]      13     15    11    13
  (0.03,2.5]        10     13    17     6
>