Home work 09.09.

The random pair (X,Y) is defined as the coordinates of the following random
point on the plane. First, we toss a fair coin. If we see heads, then we
choose the point (X,Y) uniformly from the unit square [0,1]x[0,1]. If the
outcome is tails, then we take a uniform point of [-1,0]x[-1,0].

What is the expectation of X?
What is the expectation of Y?
What is the variance of X?
What is the variance of Y?
What is the covariance of X and Y?

Reminder:
variance of X = D^2(X) = E(X^2) - (E(X))^2
covariance of X and Y = cov(X,Y) = E(XY) - (E(X))(E(Y))




Home work 09.15.

1. Consider a Markov-chain on S={1,...,9} with transition matrix

      (0 0 0 x 0 0 0 0 x)
      (0 x x 0 x 0 0 0 x)
      (0 0 0 0 0 0 0 x 0)
      (x 0 0 0 0 0 0 0 0)
   P= (0 0 0 0 x 0 0 0 0)
      (0 x 0 0 0 0 0 0 0)
      (0 x 0 0 0 x x 0 0)
      (0 0 x 0 0 0 0 0 0)
      (0 0 0 x 0 0 0 0 x)

where x signifies a positive entry, but these entries can be different.
Classify the states. Give the closed recurrent classes. Which states are
transient?

2. An inventory model:

Let I(t) be the inventory level of an item at time t. Stock levels are checked
at fixed times T_0, T_1, T_2, .... A commonly used restocking policy is that
there be two critical values of inventory s and S where 0<=s<S. If at time T_n,
the inventory level I(T_n)=:X_n is less than or equal to s, immediate
procurement is done to bring the stock level up to level S. If the stock level
X_n is in the inverval (s,S], then no replenishment is undertaken. Let D_n be
the totel demand during the time interval [T_{n-1},T_n), n=1,2,..., assume
{D_n, n>=1} is iid and independent of X_0, and suppose X_0<=S.

Let the critical values be set at s=3 and S=8, and suppose {D_n} are iid
Poisson random variables with mean (expectation) lambda=4.

(a) Show the state space of the inventory Markov-chain {X_n} is S={0,...,8},
    and give the transition matrix. (already done)
(b) If X_0=8, what is the probability that there is a shortage before the end
    of the first day (time interval)?
(c) If X_0=8, find the probability that no replenishment will be necessary
    at T_1 and T_2.
(d) Starting with X_0=4, what is the distribution of time until the first
    replenishment?




Home work 09.22.

We toss a fair coin infinitely many times. Let X_1, X_2,... be the sequence of
the outcomes. In other words, X_1, X_2,... is an iid sequence of random
variables with P(X_1=H)=P(X_1=T)=1/2. What is the expected waiting time for
the first HH, or what is E(T(HH)) if T(HH)=min{k: X_(k-1)=H, X_k=H}?