[SOLVED] Probability & statistics for eecs homework 02

1. Given n ≥ 2 numbers (a1, a2, . . . , an) with no repetitions, a bootstrap sample is a
sequence (x1, x2, . . . , xn) formed from the aj ’s by sampling with replacement with equal
probabilities. Bootstrap samples arise in a widely used statistical method known as
the bootstrap. For example, if n = 2 and (a1, a2) = (3, 1), then the possible bootstrap
samples are (3, 3),(3, 1),(1, 3), and (1, 1).(a) How many possible bootstrap samples are there for (a1, . . . , an)?
(b) How many possible bootstrap samples are there for (a1, . . . , an), if order does not
matter (in the sense that it only matters how many times each aj was chosen, not
the order in which they were chosen)?(c) One random bootstrap sample is chosen (by sampling from a1, . . . , an with replacement, as described above). Show that not all unordered bootstrap samples (in the
sense of (b)) are equally likely.Find an unordered bootstrap sample b1 that is
as likely as possible, and an unordered bootstrap sample b2 that is as unlikely as
possible. Let p1 be the probability of getting b1 and p2 be the probability of getting b2 (so pi is the probability of getting the specific unordered bootstrap sample
bi). What is p1/p2? What is the ratio of the probability of getting an unordered
bootstrap sample whose probability is p1 to the probability of getting an unordered
sample whose probability is p2?2. If each box of the broad noodle of chili oil flavor contains a coupon, and there are 108
different types of coupons. Given n ≥ 200, what is the probability that buying n boxes
can collect all 108 types of coupons? You need to plot a figure (you do NOT need
to submit the code, if used, this time) to show how such probability changes with the
increasing value of n. When such probability is no less than 95%, what is the minimum
number of n?3. A batch of one hundred garage kits is inspected by testing four randomly selected ones.
If one of the four is defective, the batch is rejected. What is the probability that the
batch is accepted if it contains five defectives?4. There are three boxes:
a. A box containing two gold coins;
b. A box containing two silver coins;c. A box containing one gold coin and a silver coin.
After choosing a box randomly and withdrawing one coin randomly, if that happens
to be a gold coin, find the probability of the next coin drawn from the same box also
being a gold coin.5. Mirana is about to play a two-game Starcraft match with an opponent, and wants to
find the strategy that maximizes his winning chances. Each game ends with either a
win by one of the players, or a draw. If the score is tied at the end of the two games, the
match goes into a sudden-death mode, and the players continue to play until the first
time one of them wins a game (and the match).Mirana has two playing styles, i.e.,
timid and bold, and she can choose one of the two at will in each game, no matter what
style she chose in previous games. With timid play, she draws with probability pd > 0,
and she loses with probability (1 − pd). With bold play, she wins with probability pw,
and she loses with probability (1 − pw). Mirana will always play bold during sudden
death, but may switch style between games 1 and 2.Find the probability that Mirana wins the match for each of the following strategies:
(a) Play bold in both games 1 and 2.
(b) Play timid in both games 1 and 2.
(c) Play timid whenever she is ahead in the score, and play bold otherwise.(d) Assume that pw < 1/2, so Mirana is the worse player, regardless of the playing
style she adopts. Show that with the strategy in (c) above, and depending on the
values of pw and pd, Mirana may have a better than a 50-50 chance to win the
match. Intuitively, how do you explain this advantage?