# Statistics functions in R


# generate a bunch of Uniform(0,1) realizations

U <- runif(20000)
U
hist(U)

X <- log(U/(1-U))
hist(X)

# Let's look at the normal distribution

X <- rnorm(1000,mean=10,sd=2) # generating random values from normal distribution

hist(X,freq = F)

# dnorm()  evaluates the probability density function

x <- seq(2,18,length=200)
pdfx <- dnorm(x,mean = 10, sd = 2)
lines(pdfx ~ x, col = "blue")


# make a plot of the standard normal density with the middle 95% 
# shaded

# qnorm()  the quantile function of the normal distribution
x <- seq(-4,4,length=200)
pdfx <- dnorm(x) # default mean is 0, and default sd = 1
plot(pdfx ~ x, type="l")

lower <- qnorm(0.025)
upper <- qnorm(0.975)

x2 <- seq(lower, upper, length=100)

x_poly <- c(lower,x2,upper)
y_poly <- c(0,dnorm(x2),0)

polygon(x = x_poly,
        y = y_poly,
        col = "red",
        border = NA)

# pnorm() evaluates the cumulative distribution function

# Make a z table????

zrow <- seq(0,3.5,by=0.1) # row values of z
zcol <- seq(0,0.09,by=0.01) # column values
nr <- length(zrow) # the number of rows
nc <- length(zcol) # number of columns
tab <- matrix(0,nrow=nr,ncol=nc) # set up an empty table

for(i in 1:nr)
  for(j in 1:nc){
    
    z <- zrow[i] + zcol[j]
    tab[i,j] <- 1 - pnorm(z)
    
  }

tab <- round(tab,4)
colnames(tab) <- zcol
rownames(tab) <- zrow
tab



# binomial distribution
# rbinom, dbinom, pbinom, and qbinom
# For the number of successes in n independent Bernoulli trials


n <- 50
p <- 1/2
# generate a single realization
rbinom(1,size = n,prob = p) # size is how many Bernoulli trials

N <- 2000

# generate N realizations
X <- rbinom(N,size = n, prob = p)

# summarize the sample:
table(X) # tells me how many times each number was observed
plot(table(X)) # this is good for discrete data.