set.seed(2021)
runif(3)
set.seed(2021)
runif(3)
runif(3)

p = 0.5;
U = runif(1, min = 0, max = 1);
X = as.numeric(U < p);
X

p = 0.2;
n = 50;
U = runif(n, min = 0, max = 1);
X = sum(U < p);
X


p1=0.35
p2=0.15
p3=0.4
p4=0.1
P = c(p1,p1+p2, p1+p2+p3, p1+p2+p3+p4)
X = c(1, 2, 3, 4)
counter = 1
set.seed(2021)
r = runif(1, min = 0, max = 1);
while(r > P[counter])
  counter = counter + 1
end
X[counter]


p1=0.35
p2=0.15
p3=0.4
p4=0.1
P = c(p1,p1+p2, p1+p2+p3, p1+p2+p3+p4)
X = c(1, 2, 3, 4)
counter = 1;
set.seed(2021)
r = runif(1, min = 0, max = 1);
for (value in P[counter]) {
  if(r > P[counter]){
    counter = counter +1
  }
}
X[counter]


U = runif(1, min = 0, max = 1);
lambda = 1
X = -(1/lambda)*log(U)


n = 20;
lambda = 10;
X = (-1/lambda)*sum(log(runif(n,min=0,max=1)))

dnorm(0)
dnorm(1)
dnorm(0, mean=4, sd=3)

pnorm(0)
pnorm(1)

x = seq(-3,3, by=0.1); y = pnorm(x) ;plot(x,y, type="l")

pnorm(0, lower.tail=FALSE)
pnorm(1, lower.tail=FALSE)

x = seq(-3,3, by=0.1); y = pnorm(x, lower.tail=FALSE)
plot(x,y, type="l")

qnorm(0.68); qnorm(0.95);qnorm(0.997)

x = seq(0,1, by=0.05); y = qnorm(x);plot(x,y, type="l")


x=rnorm(1000)
hist(x)

pnorm(1) - pnorm(-1) # within one standard deviation
pnorm(2) - pnorm(-2) # within two standard deviation
pnorm(3) - pnorm(-3) # within three standard deviation

DiceR = read.csv("Dice.csv", header=T)
names(DiceR)= "Sum"
DiceR$Sum = as.numeric(DiceR$Sum)
Dice = na.omit(DiceR$Sum)
hist(Dice)
summary(Dice)
sd(Dice)
cs = (Dice-mean(Dice))/(sd(Dice))
mean(cs)
sd(cs)
onesdcs = cs[cs >-1& cs <1] # within one sd
twosdcs = cs[cs >-2& cs <2] # within two sd
threesdcs = cs[cs >-3& cs <3]# within three sd
length(onesdcs)/length(cs)
length(twosdcs)/length(cs)
length(threesdcs)/length(cs)
library(moments)
Kurtosis = kurtosis(Dice)
Kurtosis
Skewness = skewness(Dice)
Skewness
normaldata = replicate(1000, rnorm(97,0,1),
                       simplify=FALSE)
library(moments)
mean(sapply(normaldata, kurtosis))
mean(sapply(normaldata, skewness))
normaldata = replicate(1000, rnorm(97,0,1),
                       simplify=FALSE)
library(moments)
hist(sapply(normaldata, kurtosis))

x=rnorm(1000)
qqnorm(x)
qqline(x)

qqnorm(Dice)
qqline(Dice)