Question

In this project, we want to estimate the value of it through random sampling. First, consider the unit square S = {(x, y)0 SX S1, 0 Sy s 1} and the circle C = {(x, y)/(x-1/2)2+(y-1/2)2 5 1/4} which resides inside the unit square. Suppose that we sample n points from the unit square S, uniformly and independently. Let Xi be 1 if the ith sample falls into the circle C and 0 otherwise. Define Sn = X1 + ... + Xn. (a) What is the PMF of Xi? (b) Calculate the expectation of Xi. (c) Calculate the mean and variance of Sn/n. What happens as n grows to infinity? Discuss what it means (d) Continuing from part (c), let u = E[Sn/n] 1/4. The strong law of large numbers says that Sn/n converges almost surely to u as n → 00. Using this fact, devise an estimator of the value of u. (e) By using the Chebyshev Inequality in (1), calculate the number of samples n that guarantees (epsilon, 8)-approximation of it, where = 0.01 and 8 0.01. You may want to use the fact p(1 – p) s 0.25 for p E [0, 1]. (f) Implement the estimator in (d) by writing a computer program (using any language including C/C++/Java/MATLAB/Python/etc.) (g) Does your estimator approach the true value of n as the number of samples n grows? Plot the behavior of your estimator versus n and discuss the result. =

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I need only (d) (e) (g)! !!!

Answer 1

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I not understand g

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