LAB_1.R

#NAPLES
#QUESTION 1
#Part A

r<-rep(0,100)
r[1]=13
a=17
b=0
d=100

for(i in 1:100)
  {
  r[i+1]<-(a*r[i]+b)%%d
  }
  r

##   [1] 13 21 57 69 73 41 97 49 33 61 37 29 93 81 77  9 53  1 17 89 13 21 57
##  [24] 69 73 41 97 49 33 61 37 29 93 81 77  9 53  1 17 89 13 21 57 69 73 41
##  [47] 97 49 33 61 37 29 93 81 77  9 53  1 17 89 13 21 57 69 73 41 97 49 33
##  [70] 61 37 29 93 81 77  9 53  1 17 89 13 21 57 69 73 41 97 49 33 61 37 29
##  [93] 93 81 77  9 53  1 17 89 13

hist(r,xlab= 'random number',main = "Histogram of R")

#QUESTION 1
#PART B
set.seed(0914)
r[1]<-sample(0:100, 1)
r[1]

## [1] 6

for(i in 1:100)
{
  r[i+1]<-(a*r[i]+b)%%d
}
r

##   [1]  6  2 34 78 26 42 14 38 46 82 94 98 66 22 74 58 86 62 54 18  6  2 34
##  [24] 78 26 42 14 38 46 82 94 98 66 22 74 58 86 62 54 18  6  2 34 78 26 42
##  [47] 14 38 46 82 94 98 66 22 74 58 86 62 54 18  6  2 34 78 26 42 14 38 46
##  [70] 82 94 98 66 22 74 58 86 62 54 18  6  2 34 78 26 42 14 38 46 82 94 98
##  [93] 66 22 74 58 86 62 54 18  6

hist(r, xlab= 'random number', main = "Histogram of R with seed")

#QUESTION 2
#Part A

#There is only 64 points on the graph and I believe this is because
#the period is 64, so there are 64 points.
r<-rep(0,100)
a=13
b=1
d=64

for(i in 1:100)
{
  r[i+1]<-(a*r[i]+b)%%d
}
r

##   [1]  0  1 14 55 12 29 58 51 24 57 38 47 36 21 18 43 48 49 62 39 60 13 42
##  [24] 35  8 41 22 31 20  5  2 27 32 33 46 23 44 61 26 19 56 25  6 15  4 53
##  [47] 50 11 16 17 30  7 28 45 10  3 40  9 54 63 52 37 34 59  0  1 14 55 12
##  [70] 29 58 51 24 57 38 47 36 21 18 43 48 49 62 39 60 13 42 35  8 41 22 31
##  [93] 20  5  2 27 32 33 46 23 44

plot(r[1:100],r[2:101],xlab="R [i]", ylab= "R [ i+1 ]", main= "R [ i ] against R [ i + 1 ]")

#QUESTION 2
#Part B

r<-rep(0,100)
a=13
b=0914
d=64

for(i in 1:100)
{
  r[i+1]<-(a*r[i]+b)%%d
}
r

##   [1]  0 18 60 30 24 10 20 22 48  2 44 14  8 58  4  6 32 50 28 62 56 42 52
##  [24] 54 16 34 12 46 40 26 36 38  0 18 60 30 24 10 20 22 48  2 44 14  8 58
##  [47]  4  6 32 50 28 62 56 42 52 54 16 34 12 46 40 26 36 38  0 18 60 30 24
##  [70] 10 20 22 48  2 44 14  8 58  4  6 32 50 28 62 56 42 52 54 16 34 12 46
##  [93] 40 26 36 38  0 18 60 30 24

plot(r[1:100],r[2:101],xlab="R [i]", ylab= "R [ i+1 ]", main= "R [ i ] against R [ i + 1 ] with shift")

#The period for the new graph is 32 since we had a shift.