HOMEWORK 1: SOLUTIONS
You toss a coin, independently from toss to toss, whose probability of heads is
p and of tails q = 1 p. Find the expected number of tosses required to get the
Let Xi be the outcome (H or T) at the i-th toss. Le
HOMEWORK 2: SOLUTIONS
Consider three events A+ , A , A0 in the same probability space with a probability P on it. Suppose that A+ and A are conditionally independent given
A0 . Show that P (A+ |A0 A ) = P (A+ |A0 ).
By denition, A+ and A are
HOMEWORK 3: SOLUTIONS
Consider a Markov chain whose transition diagram is as below:
(i) Which (if any) states are inessential?
(ii) Which (if any) states are absorbing?
(iii) Find the communicating classes.
(iv) Is the chain irr
HOMEWORK 5: SOLUTIONS
A process moves on the integers 1, 2, 3, 4, and 5. It starts at 1 and, on each successive
step, moves to an integer greater than its present position, moving with equal probability to each of the remaining larger integers. State v
HOMEWORK 6: SOLUTIONS
The President of the United States tells person A his or her intention to run or not to
run in the next election. Then A relays the news to B, who in turn relays the message
to C, and so forth, always to some new person. We assume