# Welcome to R!

# Basics

1 + 2
2 * 3
4 / 3

# this is a comment

1:5

for (i in 1:10) {
  print(i)
}

array <- 2:7
print(array)

round(3.1)
ceiling(3.1)
floor(3.1)

factorial(3)
factorial(10)

mean(3:8)
mean(array)

sample(1:10, 3, replace=TRUE)

args(sample)
help(sample)

my_function <- function(arr) {
  for (i in arr) {
    print(i)
  }
}

sample(1:10, 11, replace=TRUE)

my_function(array)

# Sets
install.packages('sets')
library(sets)

a <- set(1, 2, 3)
b <- set(1, 2, 3, 4, 5)
c <- as.set(2:7)

print(a)
print(b)

is.set(a)

set_is_empty(b)
set_is_empty(set())

a | b # union
b - a # difference
a & b # intersection
a * b # cartesian product
a < b # subset
length(a) # cardinality 
a == b # equal
2^a # power set
set_combn(b, 3) 
mean(b)

# Probability Distributions

# prefixes
# d = PMF
# p = CMF
# r = simulations (random)

# suffixes
# binom(x, size=, p=)
# geom(x, prob=)
# norm(x, mean=, sd=)
# pois(x, lambda=)
# exp(x, rate)

# PMF of binomial measure where x = 1, n = 2, p = 0.5 
dbinom(1, size=2, prob=0.5)

# Binomial Distribution 
n <- 10
p <- 1/2
x <- 0:n
pmfbinom <- dbinom(x, size=n, prob=p)
pmfbinom 
plot(c(1, 2, 3, 4, 5), c(1, 4, 9, 16, 25))

plot(x, pmfbinom, main=sprintf("PMF Binomial Distribution, n = %s p = %s", n, p))
plot(x, pbinom(x, size=n, prob=p), main=sprintf("CMF Binomial Distribution, n = %s p = %s", n, p))

draws <- rbinom(x, size=n, prob=p)
draws

hist(draws, breaks=(0:n+1)-0.5, main=sprintf("random draws from bin(n = %s, p = %s)", n, p), probability=TRUE)
points(x, pmfbinom)

# geometric dist, prob = 1/3
args(dgeom)
plot(dgeom(0:10, prob=1/3))

# normal dist, mu = 0, sd = 1
args(dnorm)
plot(dnorm(-5:5, mean=0, sd=1))

# poisson dist, lambda = 2
args(dpois)
plot(dpois(0:10, lambda=2))

args(dexp)
plot(dexp(0:10, rate=0.5))

# Combinations and Permutations
install.packages("combinat")
library(combinat)

args(combn) # combinations of all elements of vector x taken m at a time
args(permn) # permutations of all elements of vector x 
# sample()

c <- combn(1:5, 2)
p <- permn(1:3)
length(p)
dim(c)[2]

c
p