Description
Two modest pencil-and-paper problems:
A. Suppose that a salesperson for a company is given a reward check (representing R) of $1000
when he/she completes a specified number of sales. After receiving a check, the salesperson
begins anew working toward the next reward check. Suppose that the probability density
function p() for , representing the time between reward checks, is given by p() =
{ } 3 for 0 2;2 2 3 for 2 3;0 otherwise . Considering the above as a regenerative
process, carry out parts (a) − (c) below:
(a) Plot the function p() (hand sketch is acceptable) and give the numerical value of l.
(b) Suppose that the standard regenerative estimate l
ˆ is formed based on one sample (i.e., one
regeneration period). What is the magnitude of the bias in l
ˆ as an estimator of l?
(c) Can the Kantorovich inequality (see HW #8) be used to provide an upper bound to the bias in
part (b)? If so, compute the bound; if not, explain why not.
B. Suppose we have two very simple Monte Carlo “simulations” involving single random
variables X and Y. Let U represent a U(0,1) random number. Do the following:
(a) Consider two cases for X and Y: (i) X = U 2
and Y = U 3
and (ii) X = U 2
and Y = (1 – U )
3
. For
each of the cases (i) and (ii), determine analytically whether CRNs (same U values in X and Y)
lead to a decrease or increase in var(X − Y) relative to independent sampling (independent U
values) for X and Y. Provide brief comments on the reason for your conclusions.
(b) Redo part (a) for case (ii) above, but with X and Y generated via the inverse transform
method rather than the transforms given in part (a). (Note that for a random variable X with
distribution function F(x), the random variable g(X) has distribution function F(g−1
(x)) if g is
increasing and 1 − F(g−1
(x)) if g is decreasing, where g−1
is the inverse function of g.) Why is the
result different from the result in part (a)?
