S Commands for Problem I

For the value of N decided upon in the first part of this problem, generate 100 samples of size N from the exponential distribution and store them in a matrix using the command

X <- matrix(rexp(100*N),100,N) We are now in a position to calculate values of for . For example, suppose we want to generate 100 random variables from the Poisson process with mean . First we create a matrix cum where each row of cum contains the cumulative sum of each row of X. This can be done using the apply command in S. Hence the command we would give would be

cum <- t(apply(X,1,cumsum)) In S the command t(A) transposes the matrix A. The reason for taking the transpose is that the cumsum function in S when applied to the rows of X creates columns of the cumulative values of the rows of X.

Then we create an indicator matrix, say, indicat that is of the same size as X and whose th element is 1 if X(i,j) is less than or equal to 5 and 0 otherwise. For this we use the command

indicat[cum <= 5] <- 1 Finally you get 100 values from the Poisson(5) process by the following command

Nt <- apply(indicat,1,sum) which just sums up each row of the matrix indicat. Nt now contains 100 observations of the random variable . Similarly you will get for . To compare the relative frequencies with the theoretical distribution, you will have to implement a function that generates the probability vector of a Poisson distribution.

S Help for Section II

1. Proceed as above to generate the processes using the cumsum function to get . Calculating from X can be done using the commands given in the previous section.

2. The 100 values of are simply the elements of the 30th column of X.