Question

Use linear congruential generator with seed value 31, a = 11, c = 17, m = 100 and inverse transform technique to generate:

1. 10 continuous uniform random variates on interval [1; 10];                                             [5]
2. 10 exponential random variates with parameter λ = 3.                                                     [2]
3. It is suspected that the generated random numbers (R_i) are autocorrelated, since an observer notices that every 4^th number starting (but not counting) from the 1^st number (i.e. i = 1) is larger than the numbers preceding or following it. Test the hypothesis of independence of random numbers at 2-sided level of significance α = 0.05 using autocorrelation.                                                                                                        [10]

Answer 1

#uniform u(1,10)
a=11;c=17;m=100;
n=10;seedx=31;
x=vector("numeric")
for (i in 1:n) {
x[i]=(a*seedx+c)%%m
seedx=x[i]
  
}
z=x/sum(x)
u=round(z*9,0)+1
u # uniform 1 to 10

#Exponential with lamda=3
exp.no<- function(n,lamda) {
if(lamda<=0 && n<=0)
{
print("wrong")
}
else
{
u=runif(n)
x=NULL
x1=NULL
for (i in 1:n) {
x=-lamda*log(1-u[i])
x1=append(x1,x,after = length(x))
}
}
return(x1)
}
a=exp.no(10,3) # exp(n=10,lamda=3)
#install.packages("randtests)
library(randtests)
randtests::runs.test(a,alternative = "two.sided")