HOMEWORK 1: SOLUTIONS
1.
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
rst head.
Solution.
Let Xi be the outcome (H or T) at the i-th toss. Le

HOMEWORK 2: SOLUTIONS
1.
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 ).
Solution.
By denition, A+ and A are

HOMEWORK 3: SOLUTIONS
1.
Consider a Markov chain whose transition diagram is as below:
(i) Which (if any) states are inessential?
1
0.3
1
2
3
(ii) Which (if any) states are absorbing?
(iii) Find the communicating classes.
0.7
1
1
0.2
(iv) Is the chain irr

HOMEWORK 5: SOLUTIONS
1.
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
1.
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