ISE 538 Midterm 10/2021: All parts are worth 10 points.
1. B’s team will play a total of 4 games this year. B will play in each game
with probability .6, independently from game to game. The number of
goals scored by B in a game she plays is Poisson with mean 2. Let X be
the total number of goals that B scores this year.
(a) Find E[X].
(b) If B’s team wins each game that she plays with probability .8 and wins
each game she does not play with probability .5, what is the expected
number of games that her team wins.
2. There are 2 coins, with coin i landing heads with probability pi, where
p1 = .8, p2 = .4. At each stage one of the coins is flipped. If it lands
heads, then that coin is used for the next flip; if tails, then the other coin
is used.
(a) Let Xn be the coin used on flip n. Is Xn, n ≥ 1 a Markov chain. If so,
give its transition probabilities.
(b) Let Yn be 1 if flip n lands heads, and let it be 0 if it lands tails. Is
Yn, n ≥ 1 a Markov chain. If so, give its transition probabilities.
(c) What proportion of flips land heads.
(d) What proportion of flips use coin 1.
3. X0, X1, . . . , is a sequence of independent and identically distributed ran-
dom variables with mass function pj = P (Xn = j), j ≥ 1. Let N =
min(n ≥ 1 : Xn 6= X0). For instance, if the sequence begins 5, 5, 5, 8 then
N = 3.
(a) Find E[N ]
(b) Find P (N > 5).
4. J plays a new game every day. Whether J wins the next game depends on
previous games only through how many of the last 2 games she has won.
If she has won i of her last 2 games, then she wins the next game with
probability pi, i = 0, 1, 2.
(a) Define a Markov chain that can be used to analyze the preceding.
(b) Give its transition probability matrix.