• random number generation
• simulation
• monte carlo integration

Random number generarion

Psudorandom sequence

• To recreate the same random numbers, use the function set.seed() , and we will have the psudorandom sequence

linear congruential generator (LCG)

• an algorithm that produces a swquwnce od pseudorandom numbers based on the recurrence relation formula

  • $X_n = (aX_{n-1}+c)$ ; mod $m$
  • example code

  # modular arithmatic
  3 %% 4
  # 3
  
  3 %% 3
  # 0
  
  5 %% 4
  # 0
  
  # a === b (mod m)
  # if that exists k belong to Z,
  # a - b = Kn
  
  # simulate numbers from 0,1,2,...,15
  seed <- 10
  new.random <- function(a=5, c=12, m=16){
  	out <- (a*seed + c) %% m
  	seed <<- out
  	return(out)
  }
  
  # random number generation
  
  out.length <-20
  variants <- rep(NA, out.length)
  for(i in 1:out.length){
  	variants[i] <- new.reandom()
  }
  variants
  

  

  <aside> 💡 <<- allows you to addign a new global variable in a local environment

  </aside>

  exercise

  # define a new function called my_unif()
  # simulate n ind. draws from uni(0,1)
  # draw 10,000 cases and plot a  hist
  
  my_unif() <- function(a, c, m, n){
  	out.length <-20
  	variants <- rep(NA, out.length)
  	for(i in 1:out.length){
  		variants[i] <- new.reandom(a = 1664545, c = 1013904223, m = 2^32)
  	}
  return(variants)
  }
  
  seed <- 10
  my_unif(n = 10)
  
  X <- my_unif(n = 10000)
  hist(X, prob = T)
  
  X1 <- my_unif(n = 10000)
  X2 <- my_unif(n = 10000)
  Y <- X1+X2
  hist(Y, prob = T)
  # triangle distribution
  
  # simulate rooling a fair 6-sided die
  Z <- floor(6*my_unif(n = 10000))+1
  # the probability of each number shows up
  table(Z)/length(Z)
  
  

  

my_dics_unif <- function(n = 1){
	out.length <- n
variants <- rep(Na, out.length)
}

Simulation

• Simulation from probability distributions
  • common distributions: use built-in R func (normal, gammam, poisson ext.)
  • uncommon distributions: need to use simulation
    • discrete random variables: often can use sample()
    • continuous random variables: inverse-trans

common distributions