Homework - Reliable Intuition?

Consider a renewal process. Let X be the inter renewal times; and let I and R be the length of an interval interrupted at random and its remainder, respectively. The following BASIC simulation calculates the average values of X, I, and R (based on 10,000 replications of I and R, where T is a random interruption point).

100         FOR j=1 TO 10000

110         S=0

120         T = -1000*LOG(1-RND)

130         X=

140         c=c+1

150         SX=SX+X

160         S=S+X

170         IF S

180         R=S-T: I=X

190         SR=SR+R: SI=SI+I

200         NEXT j

210         PRINT SX/c, SI/10000, SR/10000

a. Run the simulation for the case when X is exponentially distributed (that is, the renewal process is a Poisson process) with E(X) = 1.  Fill in the following table.

Comment on the assertion:  "It is intuitively obvious that E(I) = E(X) and E(R) = E(X) / 2."

b. Let X = Y + u, where u is a constant (to be treated as a parameter), and Y is a random variable with probability distribution: P(Y=1) = 0.9, P(Y=11) = 0.1. Run the simulation and fill in the values indicated in the table. Calculate the theoretical values of E(I) and E(R) according to the formulas (to be derived later): E(I) = E(X) + V(X) / E(X) and E(R) = E(I)/2.

Draw the theoretical graph of E(R) versus u. On the same graph, plot the points produced by the simulation.

Comment on the assertion: "It is intuitively obvious that as the average length of the random interval X increases (that is, as u increases), the average lengths of the interrupted interval I and its remainder R will also increase."