Question

Consider the problem of estimating π using a unit square centered at (1/2, 1/2) and an inscribed circle inside the square. We will estimate π by simulating n darts. For the nth dart, if the dart is inside the circle, then we return In = 1; otherwise, we return In = 0.

1. Are I1, I2, · · · , In independent? Under what assumption?

2. Are I1, I2, · · · , In identically distributed?

3. Let p represent the probability that a dart falls inside the circle. Calculate E[I] and Var(I). Your answers should be functions of p only.

4. Let ¯I = (1/n) Pn i=1 Ii . Calculate E[ ¯I] and Var( ¯I). Your answers should be functions of p and/or n only.

5. In class we learned that π = 4p and ˆπ = 4ˆp = 4¯I. Calculate E[ˆπ] and Var(piˆ ).

6. For large n, is ˆπ approximately normal? Why or why not?

7. Give an approximate 95% CI for π for a large n. Your answers should be a function of ¯I only.

8. For n = 100, 1000, 10000, report ˆπ and the 95% confidence interval for π.

9. Suppose that we want to make the half-width of the 95% CI for π no more than 0.01. Approimxately how many darts would be needed? Calculate it without running simulation.

I am pretty lost for this very first homework assignment for this class. The content required to solve it is either not covered in class yet, or is taught in a pre-requisite class that I have almost completely forgotten about. Please give me a hand! Any help would be appreciated!

Homework 1: Due by 1:30pm 1/17 Thursday Consider the problem of estimating π using a unit square centered at (1/2,1/2) and an inscribed circle inside the square. We will estimate π by simulating n darts. For the nth dart, if the dart is inside the circle, then we return In1; otherwise, we return In0 1. Are I, I2,., In independent? Under what assumption? 2. Are I, I2,.,In identically distributed? 3. Let p represent the probability that a dart falls inside the circle. Calculate E/] and Var(I) Your answers should be functions of p only. 4. Let 1 = (1/n)E:·山. calculate E[1] and Var(1). Your answers should be functions of p and/or n only. 5. În class we learned that π-4p and π-4p-4. Calculate E[π] and Var(pi). 6. For large n, is π approximately normal? Why or why not? 7. Give an approximate 95% CI for π for a large n. Your answers should be a function of 1 8. For n-100, 1000, 10000, report and the 95% confidence interval for . 9. Suppose that we want to make the half-width of the 95% CI for π no more than 0.01. Approimxately how many darts would be needed? Calculate it without running simula tion

Answer 1

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