1. 1.(2 pt.)The aptly-named Average Hotel has 100 rooms, each belonging to one of 100 guests. After an evening soiree, all of the guests (not thinking straight) randomly select a room to sleep in for that night. Multiple guests might end up in the same room.

   (a) (1 pt.) What is the expected number of guests that end up returning to their own hotel room?

   [We are expecting: A number, and please show your work.]

   (b) (1 pt.) What is the expected number of guests that end up in a room with exactly one other person? (Hint: you may find it easier to count by rooms instead of guests.)
   [We are expecting a mathematical expression like(6)(8)or a number like48, and please show your work.]

   2.(3 pt.)LetUkbe the universe of all strings consisting ofknumeric digits. (0000,0123, and9898are part of universeU4butb000,012,9!9!are not.) Letuidenote theithdigit of a stringu?Ukwhere 0=i

   LetHkbe a family ofkhash functions mapping universeUkto values{0,1,2, . . . ,9}whereh0?Hk hashes all strings according to their first digit. (For all stringsuwhereu0= 0,h0(u) = 0; for all strings uwhereu0= 1,h0(u) = 1; for all stringsuwhereu0= 9,h0(u) = 9.) Likewise,h1?Hkhashes all strings according to their second digit. Generally, for all stringsu?Ukwhereui=x,hi(u) =xfor 0=i

   1. (a) (1 pt.) What is an example of a maximally-sized subsetU3such thatH3is universal for the subset?

      [We are expecting: An example subset.]

   2. (b) (2 pt.) WouldHkbe a good family of hash functions (where “good” is defined as universal) to useforUkfork=3?
      [We are expecting: A short explanation (2-3 sentences) that answers the question.]

2. 3.(Plagiarism detection) (5 pt.)Hash functions are extremely good at what they do. Unsurprisingly, there are many fancier data structures that can be built on top of them. In this problem we will motivate and explore the idea of a “Bloom Filter,” which is one example of a fancier structure built on top of hash functions.

   Suppose you are hired by someone to make a plagiarism detection software for internal use so as to avoid any potentially embarrassing allegations of plagiarism. Specifically, your goal is to make a lightweight (i.e. fast, and relatively low-memory) piece of software that will take a sentence and output one of the following messages: 1) “potentially problematic, please rewrite”, or 2) “fresh like an ocean breeze.” Suppose your goal is the following: if the input sentence is something that you have already seen, you output “potentially problematic” (with probability 1), and if the input is something new, you want to output “fresh” with probability at least 0.99 (its alright if you have a few false-alarms).

1. (a) (1 pt.) First, you decide to use a hash table. You will make a has table that maps a piece of text to a bucket, then scrape the web for all English sentences, and hash each one to your table. Given a new sentence, you will check to see if it hashes to an empty bucket—if so, you will output option “fresh” otherwise you will output “potential plagiarism.” Suppose there are 1 billion unique sentences online. How many buckets will your hash table need to have to have the desired functionality?

   [We are expecting: A number (to the nearest order of magnitude) and one to two sentences of justification.]

2. (b) (2 pt.) You decide that is a little too much space usage, and consider the following approach: you choose 10 hash functions,h1,…,h10that each map sentences to the numbers 1 though 10 billion. You initialize an arrayAof 10 billion bits, initially set to 0. For each sentencesthat you encounter, you computeh1(s),h2(s),…,h10(s), and set the corresponding indices ofAto be 1 (namely you setA[h1(s)]?1, A[h2(s)]?1, . . .). Argue that after processing the 1 billion unique sentences, you expect a (1-1/(10 billion))10 billion˜0.37 fraction of the elements to be 0.

   For this part, feel free to assume that thehiare “idealized” hash functions that map each keys to a uniformly random bucket.
   [We are expecting: A paragraph with your argument.]

(c) (2 pt.) Now, given a sentences, to check if it might be plagiarized, you compute the 10 hashes of s, and check ifA[h1(s)] =A[h2(s)] =. . .=A[h10(s)] = 1. If so, you output “potential problem,” otherwise you output “fresh.” Prove that ifsis actually in your set of 1 Billion sentences, that you will output “potential problem” with probability 1, and that ifsisnotin your set of 1 Billion sentences, you will output “fresh” with probability˜1-(1-0.37)10˜0.99.

Again, feel free to assume that the hash functions are “idealized,” and that the claim of the previous part holds, namely that after processing the 1 Billion sentences, there are 3.7 billion indices in the arrayAwith value 0.
[We are expecting: Informal mathematical justifications for each of the bounds.]

 

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#1 A rise in the expected future exchange rate shifts the demand for domestic assets to the ____and causes the domestic currency to____. A) left; depreciate B) right; appreciate C) left; appreciate D) right; depreciate #2 A rise in the expected import demand shifts the demand for domestic assets to the ___ and causes the domestic currency to ___. A) left; appreciate B) right; depreciate C) right; appreciate D) left; depreciate

 

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1. A payment schedule requires 20 payments of $10,000, the first payment to be made 15 years from today. Assuming a discount rate of 10%, what is the present value of this series of payments? (2 marks)

2. Assume the payment schedule in problem 1 changes as follows:

. the payment 17 years from today is waived, and

. a payment of $20,000 is made 18 years from today.

Assuming a discount rate of 10%, what is the present value of this series of payments? (2 marks)

3. Your friend is celebrating his 35th birthday today and wants to start saving for his anticipated retirement at age 65. He wants to be able to withdraw $10,000 from his savings account on each birthday for 10 years following his retirement; the first withdrawal will be on his 66th birthday. Your friend intends to invest his money in the local savings bank, which offers 8 per cent interest per year. He wants to make equal, annual payments on each birthday in a new savings account he will establish for his retirement fund. If he starts making these deposits on his 36th birthday and continues to make deposits until he is 65 (the last deposit will be on his 65th birthday), what amount must he deposit annually to be able to make the desired withdrawals on retirement? (3 marks)

4. Calculate an equivalent annual income for 20 years on the following income stream received over 30 years if the interest rate is 10% per annum compounded annually:

Years 1-15 $15,000pa

Years 16-20 $20,000pa

Years 21-30 $30,000pa (3 marks)

 

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1. A researcher suspected that the number of between meals snacks eaten by students in a day during final examinations might depend on the number of tests a students had to take on that day. The accompanying table shows joint probabilities, estimated from a survey

Number of tests (X)

Number of snacks(Y)

0

1

2

3

0

0.07

0.09

0.06

0.01

1

0.07

0.06

0.07

0.01

2

0.06

0.07

0.14

0.03

3

0.02

0.04

0.16

0.04

a. Find the probability distribution of X and compute the mean number of test taken by students on that day

b. Find the probability distribution of Y and compute the mean number of snacks by students on that day

c. Find and interpret the conditional probability distribution of Y given that X=3.

d. Find the covariance between X and Y

e. Are number of snacks and number of tests independent?

 

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1. A patient lives for two periods, 1 and 2. Her well-being in period 2 depends on her state of health, s = 0, in which larger numbers imply better health status, as well as some healthrelated action t = 0 which is taken in period 1, but has a health impact in period 2. The patient derives utility from two sources. First, she gets instantaneous instrumental utility in period 2 from having her health behaviour match her health state. Formally, her instrumental utility is (-|s-t|). This means that in terms of instrumental utility, it is always optimal to align the action with the state, that is to set t = s. As an example, lower values of t could represent taking health concerns more seriously, for instance by doing x-rays. Then, instrumental utility implies that a more concerning health condition calls for more serious intervention. Notice that in this specific model, the variable t does not affect the health state s (the two are independent) but the well-being. Secondly, the patients derives anticipatory utility in period 1 from her beliefs about her health condition in period 2. The patient’s initial belief is that with probability p = 0.3 her health state will be s1 = 36 and with probability 1 – p = 0.7, it will be s2 = 49. Her anticipatory utility, which depends on her expected health state given her beliefs is 22· v p · s1 + (1 – p)s2. The patient’s expected total utility in period 1, which combines expected instrumental utility in period 2, plus anticipatory utility in period 1, is thus: 22 · v p · s1 + (1 – p)s2 – p|s1 – t| – (1 – p)|s2 – t| In period 1, the patient has the option of visiting a doctor to get diagnosed. The visit is free, and will remove any doubt about her future value of s. (In other words, her beliefs 1 about p will go either to p = 0 or to p = 1). If she does not visit the doctor, she will not learn any information about s, and will keep believing that the two states are equally likely. After deciding whether to go to the doctor, and after getting the diagnosis if she does go, the patient then chooses what health action t to take. (a) Write the patient’s expected total utility in period 1 as a function of t, if she decides NOT to visit the doctor. What level of intervention t (e.g. a diet) she selects? What is her expected total utility given the optimal t? (Hint: the EU function has components with absolute value. It is advisable to draw pen and paper the EU first in order to understand the problem… ) (b) Write the patient’s expected total utility as a function of t if she visits the doctor and gets a bad diagnosis, that is p = 1, so that her future health status is s1 = 36 for sure. What level of intervention t does she choose? What is her utility given the optimal t? (c) Repeat the exercise in part (b) for the case in which the patient visits the doctor and gets a good diagnosis, i.e. p = 0, so that her future health condition is s2 = 49 for sure. (d) Write the patient’s expected total utility from deciding to visit the doctor, not knowing which diagnosis she will get. This is the weighted sum of the utilities in (b) and (c), with the weights equal to the probabilities of the two possible diagnoses. Will the patient decide to visit the doctor? (e) Now suppose that the patient’s possible negative diagnosis is extreme sickness, that is s1 = 10. The other possibility is still s2 = 49, with the two health states still being determined by p and 1 – p. Using the same steps as in parts (a) through (d), solve for whether the patient goes to the doctor. (f) Conventional wisdom says that when information is more important for making choices, a person is more likely to seek out that information. Thus, availability of information about health risks and the effect of health behaviours is an optimal policy. How does the consideration of anticipatory utility alter this conventional paradigm? Is that true in the above case? 2. Consider the model we used to explain the representativeness heuristics in class (i.e. the Freddy model) and imagine Freddy’s psychology is such that the ”urn” size is N = 10. Suppose Freddy observes quarters of performance by fund manager Helga. Helga may be a skilled, mediocre or unskilled manager. A skilled fund manager has a 3/4 chance of beating the market each quarter, a mediocre manager has a 1/2 chance of beating the market each quarter and an unskilled manager has a 1/5 chance of beating the market each quarter. Because Freddy is an avid Bloomberg subscriber, he knows these odds. Importantly, in reality the performance of managers are independent from quarter to quarter. (a) Suppose first that Freddy thinks Helga is mediocre. What does Freddy think is the probability that Helga beats the market in the first quarter? Suppose that she actually beats the market on the first quarter. What does Freddy think is the probability she does it again? Suppose that she beats the market again. What does Freddy think is the probability that she will do so a third time? (b) How do the three probabilities in part (a) relate to each other? What sort of psychological bias does this reflect? (c) Now suppose that Freddy does not know whether Helga is skilled, mediocre or unskilled. He has just observed two consecutive quarters of under performance by Helga. Can he conclude which type of manager Helga is? Can he rule out any of the three type? If not, how many additional rounds he needs to conclude something? Explain your intuition… 2 (d) How many more quarters of under performance does Freddy need to observe in order to be sure of Helga’s type? (e) Now, let assume that Freddy observes the performance of a large sample of hedge-fund managers over two quarters. The sense of the next part of the exercise is to derive what Freddy concludes about the proportion of skilled, mediocre, and unskilled managers in the population. In reality, all managers in the market are mediocre. i. Let’s compute the proportions that Freddy (and any other trader) observes. What proportion of managers will beat the market twice? What proportion will have two under-performances? What proportion will have mixed performances? . ii. Suppose Freddy though that the proportion of skilled, mediocre, and unskilled managers in the population was q˜, 1-2˜q, and q˜, respectively. What does Freddy expect should be the proportion of managers who show two above-market performances in a row? iii. Given your answers to the previous two parts, what does Freddy infer is the proportion q˜ of skilled managers in the population? Provide an intuition for your answer 3. Explain, in your own words, what is the fallacy exemplified in the below excerpt. ”Correlations between the USD price of cryptoassets are constantly fluctuating due to a variety of factors – one of the most important factors is market irrationality <…> which has an effect similar to co-movement phenomenon. The below chart displays the average correlation, in USD prices, amongst all crypto currencies. The data shows that whenever correlations between these coins reach a specific positive upper bound between 0.8 and 1.0, the trend of Bitcoin against USD tends to reverse, or at least halts the previous price action. The cumulative duration of these periods totaled 513 days, or more than one-quarter of the entire sample range, indicating that the crypto market is prone to show extreme correlations. On average, these “0.8+ correlation periods” lasted for durations of about 39 days, with an average maximum correlation of 0.901. The most recent “peak correlation period” lasted 90 days until March 14, the longest such period in crypto-history. That coincided with Bitcoin’s fall from the 6, 000 range to the 3, 000 range. This high correlation suggests that market sentiment has already found a local maximum during that period, and a trend reversal may possibly ensue. Such a price movement pattern, to some extent, may reflect both the irrational behavior of market participants and some inherent traits of a young market.”. (Binance Research – Investigating Cryptoassets Cycles )

Attachments:

BE-PS3.pdf

 

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