COMMunication
RESearch.NET

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====== PS1. week02 ======

<code>
rm(list=ls())
rnorm2 <- function(n,mean,sd){ 
  mean+sd*scale(rnorm(n)) 
}

se <- function(sample) {
  sd(sample)/sqrt(length(sample))
}

ss <- function(x) {
  sum((x-mean(x))^2)
}

################################
N.p <- 1000000
m.p <- 100
sd.p <- 10

p1 <- rnorm2(N.p, m.p, sd.p)
mean(p1)
sd(p1)

p2 <- rnorm2(N.p, m.p+20, sd.p)
mean(p2)
sd(p2)

m.p1 <- mean(p1)
sd.p1 <- sd(p1)
var(p1)

hist(p1, breaks=100, col=rgb(1,1,1,1))
abline(v=mean(p1),lwd=2)
abline(v=mean(p1)-sd(p1), lwd=2)
abline(v=mean(p1)+sd(p1), lwd=2)
abline(v=c(m.p1-2*sd.p1, m.p1+2*sd.p1), lwd=2, col='red')
abline(v=c(m.p1-3*sd.p1, m.p1+3*sd.p1), lwd=2, col='green')

# area bet black = 68%
# between red = 95%
# between green = 99%

pnorm(m.p1+sd.p1, m.p1, sd.p1)
pnorm(m.p1-sd.p1, m.p1, sd.p1)
pnorm(m.p1+sd.p1, m.p1, sd.p1) - 
  pnorm(m.p1-sd.p1, m.p1, sd.p1)

pnorm(m.p1+2*sd.p1, m.p1, sd.p1) - 
  pnorm(m.p1-2*sd.p1, m.p1, sd.p1)

pnorm(m.p1+3*sd.p1, m.p1, sd.p1) - 
  pnorm(m.p1-3*sd.p1, m.p1, sd.p1)

m.p1
sd.p1

(m.p1-m.p1)/sd.p1
((m.p1-sd.p1) - m.p1) / sd.p1
(120-100)/10 
pnorm(1)-pnorm(-1)
pnorm(2)-pnorm(-2)
pnorm(3)-pnorm(3)

1-pnorm(-2)*2
pnorm(2)-pnorm(-2)

pnorm(120, 100, 10)
pnorm(2)-pnorm(-2)

zscore <- (120-100)/10
pnorm(zscore)-pnorm(-zscore)
zscore

# reminder. 
pnorm(-1)
pnorm(1, 0, 1, lower.tail = F)
pnorm(110, 100, 10, lower.tail = F)
zscore <- (110-100)/10
pnorm(zscore, lower.tail = F)

pnorm(118, 100, 10, lower.tail = F)
pnorm(18/10, lower.tail = F)

z.p1 <- (p1-mean(p1))/sd(p1)
mean(z.p1)
round(mean(z.p1),10)
sd(z.p1)
pnorm(1.8)-pnorm(-1.8)

hist(z.p1, breaks=100, col=rgb(0,0,0,0))
abline(v=c(m.p1, -1.8, 1.8), col='red')
1-(pnorm(1.8)-pnorm(-1.8))


pnorm(1)-pnorm(-1)
1-(pnorm(-1)*2)

pnorm(2)-pnorm(-2)
1-(pnorm(-2)*2)

1-(pnorm(-3)*2)

#
hist(p1, breaks=100, col=rgb(1,1,1,1))
abline(v=mean(p1),lwd=2)
abline(v=mean(p1)-sd(p1), lwd=2)
abline(v=mean(p1)+sd(p1), lwd=2)
abline(v=c(m.p1-2*sd.p1, m.p1+2*sd.p1), lwd=2, col='red')
abline(v=c(m.p1-3*sd.p1, m.p1+3*sd.p1), lwd=2, col='green')

# 68%
a <- qnorm(.32/2)
b <- qnorm(1-.32/2)
c(a, b)
c(-1, 1)

# 95%
c <- qnorm(.05/2)
d <- qnorm(1-.05/2)
c(c, d)
c(-2,2)
# 99% 
e <- qnorm(.01/2)
f <- qnorm(1-.01/2)
c(e,f)
c(-3,3)

pnorm(b)-pnorm(a)
c(a, b)
pnorm(d)-pnorm(c)
c(c, d)
pnorm(f)-pnorm(e)
c(e, f)

qnorm(.5)
qnorm(1)


################################
hist(p1, breaks=50, col = rgb(1, 0, 0, 0.5),
     main = "histogram of p1 and p2",)
abline(v=mean(p1), col="black", lwd=3)
hist(p2, add=T, breaks=50, col=rgb(0,0,1,.5))
abline(v=mean(p2), col="violet", lwd=3)

s.size <- 10

means.temp <- c()
s1 <- sample(p1, s.size, replace = T)
mean(s1)
means.temp <- append(means.temp, mean(s1))
means.temp <- append(means.temp, mean(sample(p1, s.size, replace = T)))
means.temp <- append(means.temp, mean(sample(p1, s.size, replace = T)))
means.temp <- append(means.temp, mean(sample(p1, s.size, replace = T)))
means.temp <- append(means.temp, mean(sample(p1, s.size, replace = T)))
means.temp

iter <- 1000000
# means <- c()
means <- rep(NA, iter)
for (i in 1:iter) {
  # means <- append(means, mean(sample(p1, s.size, replace = T)))
  means[i] <- mean(sample(p1, s.size, replace = T))
}
length(means)
mean(means)
sd(means)
se.s <- sd(means)

hist(means, breaks=100, col=rgb(.1, 0, 0, .5))
abline(v=mean(means), col="red", lwd=2)

# now we want to get sd of this distribution
lo1 <- mean(means)-se.s
hi1 <- mean(means)+se.s
lo2 <- mean(means)-2*se.s
hi2 <- mean(means)+2*se.s
lo3 <- mean(means)-3*se.s
hi3 <- mean(means)+3*se.s

hist(means, 
     xlim = c(mean(means)-5*sd(means), mean(means)+10*sd(means)), 
     col = rgb(1, 0, 0, .5))
abline(v=mean(means), col="black", lwd=2)
# abline(v=mean(p2), colo='darkgreen', lwd=3)
abline(v=c(lo1, hi1, lo2, hi2, lo3, hi3), 
       col=c("green","green", "blue", "blue", "orange", "orange"), 
       lwd=2)

# meanwhile . . . .
se.s

se.z <- sqrt(var(p1)/s.size)
se.z <- c(se.z)
se.z

# sd of sample means (sd(means))
# is sqrt(var(s1)/s.size) or 
# sd(s1) / sqrt(s.size) 
# = se.s

# when iter value goes to 
# unlimited value:
# mean(means) = mean(p1) 
# and
# sd(means) = sd(p1) / sqrt(s.size)
# that is, sd(means) = se.z
# This is called CLT (Central Limit Theorem)
mean(means)
mean(p1)
sd(means)
var(p1) 
sqrt(var(p1)/s.size)
se.z

sd(means)
se.s 
se.z

# because CLT
loz1 <- mean(p1)-se.z
hiz1 <- mean(p1)+se.z
loz2 <- mean(p1)-2*se.z
hiz2 <- mean(p1)+2*se.z
loz3 <- mean(p1)-3*se.z
hiz3 <- mean(p1)+3*se.z

c(lo1, loz1)
c(lo2, loz2)
c(lo3, loz3)

c(hi1, hiz1)
c(hi2, hiz2)
c(hi3, hiz3)


hist(means, 
     xlim = c(mean(means)-5*sd(means), mean(means)+10*sd(means)), 
     col = rgb(1, 0, 0, .5))
abline(v=mean(means), col="black", lwd=2)
# abline(v=mean(p2), colo='darkgreen', lwd=3)
abline(v=c(lo1, hi1, lo2, hi2, lo3, hi3), 
       col=c("green","green", "blue", "blue", "orange", "orange"), 
       lwd=2)

round(c(lo1, hi1))
round(c(lo2, hi2))
round(c(lo3, hi3))

round(c(loz1, hiz1))
round(c(loz2, hiz2))
round(c(loz3, hiz3))

m.sample.i.got <- mean(means)+ 1.5*sd(means)
m.sample.i.got

hist(means, 
     xlim = c(mean(means)-10*sd(means), mean(means)+10*sd(means)), 
     col = rgb(1, 0, 0, .5))
abline(v=mean(means), col="black", lwd=3)
abline(v=m.sample.i.got, col='darkgreen', lwd=3)

# what is the probablity of getting
# greater than 
# m.sample.i.got?
m.sample.i.got
pnorm(m.sample.i.got, mean(means), sd(means), lower.tail = F)
pnorm(m.sample.i.got, mean(p1), se.z, lower.tail = F)

# then, what is the probabilty of getting 
# greater than m.sample.i.got and
# less than corresponding value, which is
# mean(means) - m.sample.i.got - mean(means)
# (green line)
tmp <- mean(means) - (m.sample.i.got - mean(means))
abline(v=tmp, col='green', lwd=3)
2 * pnorm(m.sample.i.got, mean(p1), sd(means), lower.tail = F)
m.sample.i.got

### one more time 
mean(p2)
sd(p2)
sample(p2, s.size)
m.sample.i.got <- mean(sample(p2, s.size))
m.sample.i.got

hist(means, 
     xlim = c(mean(means)-15*sd(means), mean(means)+15*sd(means)), 
     col = rgb(1, 0, 0, .5))
abline(v=mean(means), col="black", lwd=2)
abline(v=m.sample.i.got, col='darkgreen', lwd=2)

# what is the probablity of getting
# greater than 
# m.sample.i.got?
m.sample.i.got
pnorm(m.sample.i.got, mean(p1), sd(means), lower.tail = F)

# then, what is the probabilty of getting 
# greater than m.sample.i.got and
# less than corresponding value, which is
# mean(means) - m.sample.i.got - mean(means)
# (green line)
tmp <- mean(means) - (m.sample.i.got - mean(means))
abline(v=tmp, col='green', lwd=2)
2 * pnorm(m.sample.i.got, mean(p1), sd(means), lower.tail = F)




</code>
====== output ======
<WRAP group>
<WRAP column half>
<code>
> rm(list=ls())
> rnorm2 <- function(n,mean,sd){ 
+   mean+sd*scale(rnorm(n)) 
+ }
> 
> se <- function(sample) {
+   sd(sample)/sqrt(length(sample))
+ }
> 
> ss <- function(x) {
+   sum((x-mean(x))^2)
+ }
> 
</code>
</WRAP>
<WRAP column half>
....
</WRAP>
</WRAP>

<WRAP group>
<WRAP column half>
<code>
> ################################
> N.p <- 1000000
> m.p <- 100
> sd.p <- 10
> 
> p1 <- rnorm2(N.p, m.p, sd.p)
> mean(p1)
[1] 100
> sd(p1)
[1] 10
> 
> p2 <- rnorm2(N.p, m.p+20, sd.p)
> mean(p2)
[1] 120
> sd(p2)
[1] 10
> 
> m.p1 <- mean(p1)
> sd.p1 <- sd(p1)
> var(p1)
     [,1]
[1,]  100
> 
> hist(p1, breaks=100, col=rgb(1,1,1,1))
> abline(v=mean(p1),lwd=2)
> abline(v=mean(p1)-sd(p1), lwd=2)
> abline(v=mean(p1)+sd(p1), lwd=2)
> abline(v=c(m.p1-2*sd.p1, m.p1+2*sd.p1), lwd=2, col='red')
> abline(v=c(m.p1-3*sd.p1, m.p1+3*sd.p1), lwd=2, col='green')
> 
> # area bet black = 68%
> # between red = 95%
> # between green = 99%
> 
> pnorm(m.p1+sd.p1, m.p1, sd.p1)
[1] 0.8413447
> pnorm(m.p1-sd.p1, m.p1, sd.p1)
[1] 0.1586553
> pnorm(m.p1+sd.p1, m.p1, sd.p1) - 
+   pnorm(m.p1-sd.p1, m.p1, sd.p1)
[1] 0.6826895
> 
> pnorm(m.p1+2*sd.p1, m.p1, sd.p1) - 
+   pnorm(m.p1-2*sd.p1, m.p1, sd.p1)
[1] 0.9544997
> 
> pnorm(m.p1+3*sd.p1, m.p1, sd.p1) - 
+   pnorm(m.p1-3*sd.p1, m.p1, sd.p1)
[1] 0.9973002
> 
> m.p1
[1] 100
> sd.p1
[1] 10
> 
> (m.p1-m.p1)/sd.p1
[1] 0
> ((m.p1-sd.p1) - m.p1) / sd.p1
[1] -1
> (120-100)/10 
[1] 2
> pnorm(1)-pnorm(-1)
[1] 0.6826895
> pnorm(2)-pnorm(-2)
[1] 0.9544997
> pnorm(3)-pnorm(-3)
[1] 0.9973002
> 
> 1-pnorm(-2)*2
[1] 0.9544997
> pnorm(2)-pnorm(-2)
[1] 0.9544997
> 
> pnorm(120, 100, 10)
[1] 0.9772499
> pnorm(2)-pnorm(-2)
[1] 0.9544997
> 
</code>
</WRAP>
<WRAP column half>
....
</WRAP>
</WRAP>

<WRAP group>
<WRAP column half>
<code>
> zscore <- (120-100)/10
> pnorm(zscore)-pnorm(-zscore)
[1] 0.9544997
> zscore
[1] 2
> 
> 
> pnorm(-1)
[1] 0.1586553
> pnorm(1, 0, 1, lower.tail = F)
[1] 0.1586553
> pnorm(110, 100, 10, lower.tail = F)
[1] 0.1586553
> zscore <- (110-100)/10
> pnorm(zscore, lower.tail = F)
[1] 0.1586553
> 
> pnorm(118, 100, 10, lower.tail = F)
[1] 0.03593032
> pnorm(18/10, lower.tail = F)
[1] 0.03593032
> 
</code>
</WRAP>
<WRAP column half>
....
</WRAP>
</WRAP>

<WRAP group>
<WRAP column half>
<code>
> z.p1 <- (p1-mean(p1))/sd(p1)
> mean(z.p1)
[1] -2.319195e-17
> round(mean(z.p1),10)
[1] 0
> sd(z.p1)
[1] 1
> pnorm(1.8)-pnorm(-1.8)
[1] 0.9281394
> 
> hist(z.p1, breaks=100, col=rgb(0,0,0,0))
> abline(v=c(m.p1, -1.8, 1.8), col='red')
> 1-(pnorm(1.8)-pnorm(-1.8))
[1] 0.07186064
> 
> 
</code>
</WRAP>
<WRAP column half>
....
</WRAP>
</WRAP>

<WRAP group>
<WRAP column half>
<code>
> pnorm(1)-pnorm(-1)
[1] 0.6826895
> 1-(pnorm(-1)*2)
[1] 0.6826895
> 
> pnorm(2)-pnorm(-2)
[1] 0.9544997
> 1-(pnorm(-2)*2)
[1] 0.9544997
> 
> 1-(pnorm(-3)*2)
[1] 0.9973002
> 
</code>
</WRAP>
<WRAP column half>
....
</WRAP>
</WRAP>


<WRAP group>
<WRAP column half>
<code>
> #
> hist(p1, breaks=100, col=rgb(1,1,1,1))
> abline(v=mean(p1),lwd=2)
> abline(v=mean(p1)-sd(p1), lwd=2)
> abline(v=mean(p1)+sd(p1), lwd=2)
> abline(v=c(m.p1-2*sd.p1, m.p1+2*sd.p1), lwd=2, col='red')
> abline(v=c(m.p1-3*sd.p1, m.p1+3*sd.p1), lwd=2, col='green')
> 
> # 68%
> a <- qnorm(.32/2)
> b <- qnorm(1-.32/2)
> c(a, b)
[1] -0.9944579  0.9944579
> c(-1, 1)
[1] -1  1
> 
> # 95%
> c <- qnorm(.05/2)
> d <- qnorm(1-.05/2)
> c(c, d)
[1] -1.959964  1.959964
> c(-2,2)
[1] -2  2
> # 99% 
> e <- qnorm(.01/2)
> f <- qnorm(1-.01/2)
> c(e,f)
[1] -2.575829  2.575829
> c(-3,3)
[1] -3  3
> 
> pnorm(b)-pnorm(a)
[1] 0.68
> c(a, b)
[1] -0.9944579  0.9944579
> pnorm(d)-pnorm(c)
[1] 0.95
> c(c, d)
[1] -1.959964  1.959964
> pnorm(f)-pnorm(e)
[1] 0.99
> c(e, f)
[1] -2.575829  2.575829
> 
> qnorm(.5)
[1] 0
> qnorm(1)
[1] Inf
> 
> 
</code>
</WRAP>
<WRAP column half>
....
</WRAP>
</WRAP>


<WRAP group>
<WRAP column half>
<code>
> ################################
> hist(p1, breaks=50, col = rgb(1, 0, 0, 0.5),
+      main = "histogram of p1 and p2",)
> abline(v=mean(p1), col="black", lwd=3)
> hist(p2, add=T, breaks=50, col=rgb(0,0,1,.5))
> abline(v=mean(p2), col="violet", lwd=3)
> 
</code>

<code>
> s.size <- 10
> 
> means.temp <- c()
> s1 <- sample(p1, s.size, replace = T)
> mean(s1)
[1] 99.64273
> means.temp <- append(means.temp, mean(s1))
> means.temp <- append(means.temp, mean(sample(p1, s.size, replace = T)))
> means.temp <- append(means.temp, mean(sample(p1, s.size, replace = T)))
> means.temp <- append(means.temp, mean(sample(p1, s.size, replace = T)))
> means.temp <- append(means.temp, mean(sample(p1, s.size, replace = T)))
> means.temp
[1]  99.64273 107.15516 103.81192 103.12311 105.88372
> 
</code>
</WRAP>
<WRAP column half>
....
</WRAP>
</WRAP>


<WRAP group>
<WRAP column half>
<code>
> iter <- 1000000
> # means <- c()
> means <- rep(NA, iter)
> for (i in 1:iter) {
+   # means <- append(means, mean(sample(p1, s.size, replace = T)))
+   means[i] <- mean(sample(p1, s.size, replace = T))
+ }
> length(means)
[1] 1000000
> mean(means)
[1] 99.99544
> sd(means)
[1] 3.161886
> se.s <- sd(means)
> 
> hist(means, breaks=100, col=rgb(.1, 0, 0, .5))
> abline(v=mean(means), col="red", lwd=2)
> 
</code>
</WRAP>
<WRAP column half>
....
</WRAP>
</WRAP>


<WRAP group>
<WRAP column half>
<code>
> # now we want to get sd of this distribution
> lo1 <- mean(means)-se.s
> hi1 <- mean(means)+se.s
> lo2 <- mean(means)-2*se.s
> hi2 <- mean(means)+2*se.s
> lo3 <- mean(means)-3*se.s
> hi3 <- mean(means)+3*se.s
> 
> hist(means, 
+      xlim = c(mean(means)-5*sd(means), mean(means)+10*sd(means)), 
+      col = rgb(1, 0, 0, .5))
> abline(v=mean(means), col="black", lwd=2)
> # abline(v=mean(p2), colo='darkgreen', lwd=3)
> abline(v=c(lo1, hi1, lo2, hi2, lo3, hi3), 
+        col=c("green","green", "blue", "blue", "orange", "orange"), 
+        lwd=2)
> 
</code>
</WRAP>
<WRAP column half>
....
</WRAP>
</WRAP>


<WRAP group>
<WRAP column half>
<code>
> # meanwhile . . . .
> se.s
[1] 3.161886
> 
> se.z <- sqrt(var(p1)/s.size)
> se.z <- c(se.z)
> se.z
[1] 3.162278
> 
> # sd of sample means (sd(means))
> # = se.s
> 
> # when iter value goes to 
> # unlimited value:
> # mean(means) = mean(p1) 
> # and
> # sd(means) = sd(p1) / sqrt(s.size)
> # that is, sd(means) = se.z
> # This is called CLT (Central Limit Theorem)
> mean(means)
[1] 99.99544
> mean(p1)
[1] 100
> sd(means)
[1] 3.161886
> var(p1) 
     [,1]
[1,]  100
> sqrt(var(p1)/s.size)
         [,1]
[1,] 3.162278
> se.z
[1] 3.162278
> 
> sd(means)
[1] 3.161886
> se.s 
[1] 3.161886
> se.z
[1] 3.162278
> 
> # because CLT
> loz1 <- mean(p1)-se.z
> hiz1 <- mean(p1)+se.z
> loz2 <- mean(p1)-2*se.z
> hiz2 <- mean(p1)+2*se.z
> loz3 <- mean(p1)-3*se.z
> hiz3 <- mean(p1)+3*se.z
> 
> c(lo1, loz1)
[1] 96.83356 96.83772
> c(lo2, loz2)
[1] 93.67167 93.67544
> c(lo3, loz3)
[1] 90.50978 90.51317
> 
> c(hi1, hiz1)
[1] 103.1573 103.1623
> c(hi2, hiz2)
[1] 106.3192 106.3246
> c(hi3, hiz3)
[1] 109.4811 109.4868
> 
> 
</code>
</WRAP>
<WRAP column half>
....
</WRAP>
</WRAP>


<WRAP group>
<WRAP column half>
<code>
> hist(means, 
+      xlim = c(mean(means)-5*sd(means), mean(means)+10*sd(means)), 
+      col = rgb(1, 0, 0, .5))
> abline(v=mean(means), col="black", lwd=2)
> # abline(v=mean(p2), colo='darkgreen', lwd=3)
> abline(v=c(lo1, hi1, lo2, hi2, lo3, hi3), 
+        col=c("green","green", "blue", "blue", "orange", "orange"), 
+        lwd=2)
> 
> round(c(lo1, hi1))
[1]  97 103
> round(c(lo2, hi2))
[1]  94 106
> round(c(lo3, hi3))
[1]  91 109
> 
> round(c(loz1, hiz1))
[1]  97 103
> round(c(loz2, hiz2))
[1]  94 106
> round(c(loz3, hiz3))
[1]  91 109
> 
</code>
</WRAP>
<WRAP column half>
....
</WRAP>
</WRAP>


<WRAP group>
<WRAP column half>
<code>
> m.sample.i.got <- mean(means)+ 1.5*sd(means)
> m.sample.i.got
[1] 104.7383
> 
> hist(means, 
+      xlim = c(mean(means)-10*sd(means), mean(means)+10*sd(means)), 
+      col = rgb(1, 0, 0, .5))
> abline(v=mean(means), col="black", lwd=3)
> abline(v=m.sample.i.got, col='darkgreen', lwd=3)
> 
> # what is the probablity of getting
> # greater than 
> # m.sample.i.got?
> m.sample.i.got
[1] 104.7383
> pnorm(m.sample.i.got, mean(means), sd(means), lower.tail = F)
[1] 0.0668072
> pnorm(m.sample.i.got, mean(p1), se.z, lower.tail = F)
[1] 0.06701819
> 
> # then, what is the probabilty of getting 
> # greater than m.sample.i.got and
> # less than corresponding value, which is
> # mean(means) - m.sample.i.got - mean(means)
> # (green line)
> tmp <- mean(means) - (m.sample.i.got - mean(means))
> abline(v=tmp, col='green', lwd=3)
> 2 * pnorm(m.sample.i.got, mean(p1), sd(means), lower.tail = F)
[1] 0.1339882
> m.sample.i.got
[1] 104.7383
> 
</code>
</WRAP>
<WRAP column half>
....
</WRAP>
</WRAP>


<WRAP group>
<WRAP column half>
<code>
> ### one more time 
> mean(p2)
[1] 120
> sd(p2)
[1] 10
> sample(p2, s.size)
 [1] 120.9639 119.8341 134.0344 122.5446 121.4557 113.6820 114.7603 105.2138 122.7027 125.4131
> m.sample.i.got <- mean(sample(p2, s.size))
> m.sample.i.got
[1] 120.2492
> 
> hist(means, 
+      xlim = c(mean(means)-15*sd(means), mean(means)+15*sd(means)), 
+      col = rgb(1, 0, 0, .5))
> abline(v=mean(means), col="black", lwd=2)
> abline(v=m.sample.i.got, col='darkgreen', lwd=2)
> 
> # what is the probablity of getting
> # greater than 
> # m.sample.i.got?
> m.sample.i.got
[1] 120.2492
> pnorm(m.sample.i.got, mean(p1), sd(means), lower.tail = F)
[1] 7.560352e-11
> 
> # then, what is the probabilty of getting 
> # greater than m.sample.i.got and
> # less than corresponding value, which is
> # mean(means) - m.sample.i.got - mean(means)
> # (green line)
> tmp <- mean(means) - (m.sample.i.got - mean(means))
> abline(v=tmp, col='green', lwd=2)
> 2 * pnorm(m.sample.i.got, mean(p1), sd(means), lower.tail = F)
[1] 1.51207e-10
> 
</code>
</WRAP>
<WRAP column half>
....
</WRAP>
</WRAP>