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7. Consider a calculation of \(\pi\) by simulation with random numbers. Implement Monte Carlo scheme to estimate \(\pi\). [Random number generation with MS Excel] Test of a quarter of unit circle with its center at origin inside a square: \(0 < x < 1, 0 < y < 1\). Step 1: Generate \((x,y)\) coordinate from a pair of independent random uniform distributions: Rand() function in Excel generates random number following U(0,1) Step 2: Calculate the distance of a random point \((x,y)\) from the origin: \(x^2 + y^2 \le 1\) Step 3: Count the point \((x,y)\) if it is within the circle, else move to Step 1 \(P(\text{point is located in the circle}) = \frac{\text{# of points within the circle}}{\text{# of total points generated}} \approx \frac{\text{Area of Circle}}{\text{Area of square}} = \frac{\pi}{4}\) \(\text{Therefore, } \pi \approx 4 \frac{\text{# of points within the circle}}{\text{# of total points generated}}\) (a) Plot your estimates for sample sizes of n = 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024. (10 points) (b) What is your conclusion based on the result of (b)? (10 points)

          7. Consider a calculation of \(\pi\) by simulation with random numbers. Implement Monte Carlo scheme to
estimate \(\pi\).
[Random number generation with MS Excel]
Test of a quarter of unit circle with its center at origin inside a square: \(0 < x < 1, 0 < y < 1\).
Step 1: Generate \((x,y)\) coordinate from a pair of independent random uniform distributions: Rand()
function in Excel generates random number following U(0,1)
Step 2: Calculate the distance of a random point \((x,y)\) from the origin: \(x^2 + y^2 \le 1\)
Step 3: Count the point \((x,y)\) if it is within the circle, else move to Step 1
\(P(\text{point is located in the circle}) = \frac{\text{# of points within the circle}}{\text{# of total points generated}} \approx \frac{\text{Area of Circle}}{\text{Area of square}} = \frac{\pi}{4}\)
\(\text{Therefore, } \pi \approx 4 \frac{\text{# of points within the circle}}{\text{# of total points generated}}\)
(a) Plot your estimates for sample sizes of n = 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024. (10 points)
(b) What is your conclusion based on the result of (b)? (10 points)

7. Consider a calculation of π by simulation with random numbers. Implement Monte Carlo scheme to
estimate π.
[Random number generation with MS Excel]
Test of a quarter of unit circle with its center at origin inside a square: 0 < x < 1, 0 < y < 1.
Step 1: Generate (x,y) coordinate from a pair of independent random uniform distributions: Rand()
function in Excel generates random number following U(0,1)
Step 2: Calculate the distance of a random point (x,y) from the origin: x^2 + y^2 ≤ 1
Step 3: Count the point (x,y) if it is within the circle, else move to Step 1
P(point is located in the circle) = (# of points within the circle)/(# of total points generated)≈(Area of Circle)/(Area of square) = (π)/(4)
Therefore, π≈ 4 (# of points within the circle)/(# of total points generated)
(a) Plot your estimates for sample sizes of n = 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024. (10 points)
(b) What is your conclusion based on the result of (b)? (10 points)

Added by Paul M.

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