Problem 4 (counts as two problems): Let X1, X2, ... , X,be a collection of independent discrete random variables that all take the value 1 with probability p and take the value 0 with probability (1-p). The following set of steps illustrates the Law of Large Numbers at work. a) Compute the mean and the variance of X1 (which is the same for X2, X3, etc.) b) Use your answer to (a) to compute the mean and variance of p = (X1 + X2 + .+ X„), which is the proportion of "ones" observed in the n instances of X;. c) Suppose n= 10,000. Use Chebyshev's inequality to provide an upper bound for the probability that the difference between p and p exceeds 0.05. d) Use calculus to show that if p is a number between 0 and 1, then p(1 – p) < e) Use your answers to (c) and (d) to provide an upper bound, that does not depend on p, for the probability that the difference between p and p exceeds 0.05. f) Interpret this problem in the context of randomly sampling 10,000 people from a large population, asking them a yes-no question, and using the result to make an inference about the whole population. %3D

Question
Problem 4 (counts as two problems): Let X1, X2, ... , X,be a collection of independent discrete random variables
that all take the value 1 with probability p and take the value 0 with probability (1-p). The following set of steps
illustrates the Law of Large Numbers at work.
a) Compute the mean and the variance of X1 (which is the same for X2, X3, etc.)
b) Use your answer to (a) to compute the mean and variance of p = (X1 + X2 + .+ X„), which is the
proportion of "ones" observed in the n instances of X;.
c) Suppose n= 10,000. Use Chebyshev's inequality to provide an upper bound for the probability that the
difference between p and p exceeds 0.05.
d) Use calculus to show that if p is a number between 0 and 1, then p(1 – p) <
e) Use your answers to (c) and (d) to provide an upper bound, that does not depend on p, for the probability that
the difference between p and p exceeds 0.05.
f) Interpret this problem in the context of randomly sampling 10,000 people from a large population, asking
them a yes-no question, and using the result to make an inference about the whole population.
%3D
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Transcribed Image Text

Problem 4 (counts as two problems): Let X1, X2, ... , X,be a collection of independent discrete random variables that all take the value 1 with probability p and take the value 0 with probability (1-p). The following set of steps illustrates the Law of Large Numbers at work. a) Compute the mean and the variance of X1 (which is the same for X2, X3, etc.) b) Use your answer to (a) to compute the mean and variance of p = (X1 + X2 + .+ X„), which is the proportion of "ones" observed in the n instances of X;. c) Suppose n= 10,000. Use Chebyshev's inequality to provide an upper bound for the probability that the difference between p and p exceeds 0.05. d) Use calculus to show that if p is a number between 0 and 1, then p(1 – p) < e) Use your answers to (c) and (d) to provide an upper bound, that does not depend on p, for the probability that the difference between p and p exceeds 0.05. f) Interpret this problem in the context of randomly sampling 10,000 people from a large population, asking them a yes-no question, and using the result to make an inference about the whole population. %3D