### Illustration of the linear congruential method. pp67-68 Problem 72

# develop the algorithm to run the gnerator.

a <- 5

c <- 7

m <- 8

n <- 20 # number of random values to be generated

x <- numeric(n)

x[1] <- 4 # seed or x_0

for (i in 2:n){

x[i] <- a*x[i-1]+c

x[i] <- x[i] - trunc(x[i]/m)*m # this calculates the remainder, i.e., calculates the mod
}

x

# Question: What does m control in this random number generator? Why should the value of m be large?

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# A function that generates psudo random uniform values using the linear congruential method.

RANU <- function(seed,n) {

# a,c,m are the values need for the linear congruential method.

# seed is the value of the seed for the random number generator.

# n is the length of the vector returned.

a <- 65539

c <- 0

m <- 2**31

x <- numeric(n) # creates the output vecotor of length n

x[1] <- seed

for (i in 2:n){

x[i] <- a*x[i-1]+c

x[i] <- x[i] - trunc(x[i]/m)*m # this calculates the remainder, i.e., calculates the mod

}

x <- x/m # dividing by m gives values between [0,1)
return(x)

}

# A function that generates psudo random exponential values using the inverse cdf method.

RANDEXPO <- function(lam,seed,n){

# lam is the scale parameter in the density. f(x) = lam*exp(-lam*x) x >= 0

# seed is the value of the seed for the random number generator.

# n is the length of the vector returned.

u <- RANU(seed,n) # creates a vector of random uniform values

x <- -log(u)/lam # creates the output vecotor of length n

return(x)

}

### Illustration of the RANU.

seed <- 4

n <- 10000 # number of random values to be generated

u <- RANU(seed,n)

# plot a histogram

br <- seq(0,1,0.1)

hist(u,main='histogram of random uniform values',probability=T,breaks=br)

# make a scatter plot of pairs of values.

x <- u[1:trunc(n/2)]

y <- u[(trunc(n/2)+1):n]

plot(x,y,main='scatter plot of pairs of random uniform values')

### Illustration of RANDEXPO.

lambda <- 1

e <- RANDEXPO( lambda,seed,n)

hist(e,main='histogram of random exponential values',probability=T)

plot(e)