Math 525, Fall 2008: practice midterm exam no 2
(1) Use the law of iterated expectations in the form E[X] = E[E[X|Y]] for any random variables
Xand Yto prove the formula
Var[X] = Var[E[X|Y]] + E[Var[X|Y]].
(2) Let V1,V2, . . . be independent random variables, uniformly distributed on the interval (0,1).
Fix a number x, 0 < x < 1, and define a random variable N=N(x) by
N= min{n > 0|V1+V2+· · · +Vn> x}.
Compute the survival function of N(i.e. compute P{N > n}for n≥0).
(3) Suppose that 2 purple balls and 2 indigo balls are distributed equally among two blue
containers. At each trial, a ball is chosen randomly from each container and put in the
other container. Let Xnbe the number of purple balls in the first container after nsuch
moves.
(a) Explain why Xis a stationary Markov chain.
(b) Write down the transition matrix for X.
(c) Suppose there is currently one indigo ball in each container. What is the probability
that the first container will have no purple balls after four moves.
(4) Specify the classes of the Markov chain associated with the transition matrix Pbelow and
determine whether they are transient or recurrent.
P=
0.5 0.5 0 0 0
0.6 0.4 0 0 0
0 0 1 0 0
0 0 0.3 0.7 0
0 1 0 0 0
(5) Mr Depp is never happy two days in a row. During any day, he is either happy, ok, or sad.
If he is happy one day, then he is equally likely to be ok or sad the next day. If he is ok or
sad one day, then there is one chance in two that he will be the same the next day, and if
his mood changes, it is equally likely to be either of the other two possibilities. In the long
run, what proportion of the days is Mr Depp happy. What proportion of days is he sad?
(6) In an endurance competition, six participants try keep their balance while standing on one
leg for as long as possible. Suppose that the length of time the participants can do this
are given by independent exponential random variables with mean thirty minutes. The
competition ends when there the last participant loses his/her balance (and wins a sum of
money proportional to the square of the length of the competition).
(a) Find the expected value of the time at which the first participant loses his/her balance.
(b) Given that all competitors are still in the game after ten minutes, find the probability
that the competition will end within forty minutes.
(7) Customers arrive at a store according to a Poisson process with on average one customer
arriving every minute.
(a) Find the probability that the first customer arrives within one minute.
(b) Given that exactly one customer arrived within the first two minutes, find the proba-
bility that he/she arrived within the first minute.
(c) Given that exactly two customers arrived within the first two minutes, find the prob-
ability that the first customer arrived within the first minute.