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1) What is the distribution of N * Pn?

1) What is the distribution of N * Pn?

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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Problem 1: Asking Embarrassing Questions, Politely When doing polling, for instace to figure out how popular a given candidate is, a common trick is to just ask N many people whether they support that candidate, and take the support to be the faction of people who say yes: if 70 people support the candidate out of 100 asked, we estimate the support at 70% or 0.7. Suppose that the probability a person supports a candidate is p, which you do not know. Let PN be the fraction of N people polled who support the candidate: total supporters divided by N people polled. 1) What is the distribution of N * p,? 2) Show hat the expected value of pN is p, i.e., pN is a valid estimator for p. If you want your estimated value of p to be accurate, you want your 'error' on pN to be small. 3) How many people N should you poll to guarantee the erpected squared error on pN is less than e? 4) How many people N should you poll to guarantee the expected squared error on py is less than e, even if you don't know p? In the previous to problems, we considered the average or expected squared error. But just because the expected error is small doesn't mean the actual error is small. 5) How many people N should you poll to guarantee the actual error on PN is less than e with 95% confidence, even if you don't know p? However, a problem with polling is whether or not people are willing to answer honestly. If a question might be viewed as shameful or embarrassing (about politics, sexual activity, or whatever people are sensitive about), they may be reluctant to answer honestly. A potential solution to this is the following: when asking whether they support or do not support a given candidate, give the people you are polling the following instructions: flip a fair coin privately, and if it comes up HEADS, answer honestly; if it comes up TAILS, flip another fair coin and if it comes up HEADS, answer 'support', if it comes up TAILS, answer do not 'support'. In this case, the person being polled can always claim that whatever they answered was the result of the coin in a sense, the results are anonymized and the people being polled are protected. Let p be the probability that a randomly polled person using this method says 'support'; let q be the true probability a random person actually supports the candidate. We would like to know the value of q, but we can only estimate the value of p: let p, be the fraction of N people who answer 'support' using this method. We have that ElPN] = p, as before. 7) What is the relationship between q and p? 8) Construct an estimator qy from PN (i.e., a formula for ĝn in terms of pN) so that the expected value ElâN] = q. What is the variance of ĝn? 9) How many people N should you poll to guarantee the actual error between qn and q is less than e, with 90% confidence? Note, q is unknown, so you cannot use it to determine N.
Transcribed Image Text:Problem 1: Asking Embarrassing Questions, Politely When doing polling, for instace to figure out how popular a given candidate is, a common trick is to just ask N many people whether they support that candidate, and take the support to be the faction of people who say yes: if 70 people support the candidate out of 100 asked, we estimate the support at 70% or 0.7. Suppose that the probability a person supports a candidate is p, which you do not know. Let PN be the fraction of N people polled who support the candidate: total supporters divided by N people polled. 1) What is the distribution of N * p,? 2) Show hat the expected value of pN is p, i.e., pN is a valid estimator for p. If you want your estimated value of p to be accurate, you want your 'error' on pN to be small. 3) How many people N should you poll to guarantee the erpected squared error on pN is less than e? 4) How many people N should you poll to guarantee the expected squared error on py is less than e, even if you don't know p? In the previous to problems, we considered the average or expected squared error. But just because the expected error is small doesn't mean the actual error is small. 5) How many people N should you poll to guarantee the actual error on PN is less than e with 95% confidence, even if you don't know p? However, a problem with polling is whether or not people are willing to answer honestly. If a question might be viewed as shameful or embarrassing (about politics, sexual activity, or whatever people are sensitive about), they may be reluctant to answer honestly. A potential solution to this is the following: when asking whether they support or do not support a given candidate, give the people you are polling the following instructions: flip a fair coin privately, and if it comes up HEADS, answer honestly; if it comes up TAILS, flip another fair coin and if it comes up HEADS, answer 'support', if it comes up TAILS, answer do not 'support'. In this case, the person being polled can always claim that whatever they answered was the result of the coin in a sense, the results are anonymized and the people being polled are protected. Let p be the probability that a randomly polled person using this method says 'support'; let q be the true probability a random person actually supports the candidate. We would like to know the value of q, but we can only estimate the value of p: let p, be the fraction of N people who answer 'support' using this method. We have that ElPN] = p, as before. 7) What is the relationship between q and p? 8) Construct an estimator qy from PN (i.e., a formula for ĝn in terms of pN) so that the expected value ElâN] = q. What is the variance of ĝn? 9) How many people N should you poll to guarantee the actual error between qn and q is less than e, with 90% confidence? Note, q is unknown, so you cannot use it to determine N.
Let us suppose that instead of one candidate, there are M many candidates, each with probability q1, 42, 93, ..., qM of support (with q1 + q2 + ... + qM = 1). We can generalize the above polling method in the following way: each person polled flips a coin; if it comes up heads, they name the candidate they support honestly; if it comes up tails, they pick a candidate 1, 2, 3, ..., M uniformly at random and claim they support them. Let p; be the probability that a person polled claims they support candidate i, and py be the fraction of N people polled who claim they support candidate i. Again, we would like to use the py to estimate qi. 10) Build a set of estimators ĝk, đN ..., N from pN, PN 1., PN, so that E = q; for each i. 11) What is the total expected squared error of the q-estimators? i.e., what is (aN – 4i)|? (1) Bonus Security researchers frequently would like to know the probability people pick things for their 4-digit PINS (how often do people lock their phones with just 1234?). If you just ask people what PIN they use, they either will not tell you or will lie. People may not even want to use something like the strategy in this problem, because there's some probability that they may be asked to just give their PIN honestly. How could you build a polling strategy that could successfully estimate the probabilities people use various PINS with, but wouldn't require the person to ever give up their PIN entirely and clearly?
Transcribed Image Text:Let us suppose that instead of one candidate, there are M many candidates, each with probability q1, 42, 93, ..., qM of support (with q1 + q2 + ... + qM = 1). We can generalize the above polling method in the following way: each person polled flips a coin; if it comes up heads, they name the candidate they support honestly; if it comes up tails, they pick a candidate 1, 2, 3, ..., M uniformly at random and claim they support them. Let p; be the probability that a person polled claims they support candidate i, and py be the fraction of N people polled who claim they support candidate i. Again, we would like to use the py to estimate qi. 10) Build a set of estimators ĝk, đN ..., N from pN, PN 1., PN, so that E = q; for each i. 11) What is the total expected squared error of the q-estimators? i.e., what is (aN – 4i)|? (1) Bonus Security researchers frequently would like to know the probability people pick things for their 4-digit PINS (how often do people lock their phones with just 1234?). If you just ask people what PIN they use, they either will not tell you or will lie. People may not even want to use something like the strategy in this problem, because there's some probability that they may be asked to just give their PIN honestly. How could you build a polling strategy that could successfully estimate the probabilities people use various PINS with, but wouldn't require the person to ever give up their PIN entirely and clearly?
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