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A Brief Look at Random Numbers

A Brief Look at Random Numbers. CSCI 317 Mike Heroux. Computing Estimate of Pi. Area of circle C is A C = π r 2 . Area of square S is A S = s 2 . Pick random (x,y) pairs with x and y between (-1/2, 1/2). Probability of (x, y) being inside C is A C / A S .

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A Brief Look at Random Numbers

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  1. A Brief Look at Random Numbers CSCI 317 Mike Heroux CSCI 317 Mike Heroux

  2. Computing Estimate of Pi • Area of circle C is AC = πr2. • Area of square S is AS = s2. • Pick random (x,y) pairs with x and y between (-1/2, 1/2). • Probability of (x, y) being inside C is AC / AS. • In our case: • AC = π(1/2)2 = π/4. • AS = 12 = 1. • AC / AS = π/4. Circle Radius r=1/2 (x, y) CSCI 317 Mike Heroux

  3. Algorithm to Estimate Pi • Get ntotal number random pairs (x, y). • For each pair: • Compute distance from center of circle: d = sqrt(x*x+y*y). • If d < radius of circle (d<1/2) increase count of points inside circle (ninside ) by 1. • Theory tells us: ninside / ntotal≈ π/4. • Multiplying both sides above by 4, we have: π ≈ 4* (ninside / ntotal ). CSCI 317 Mike Heroux

  4. Matlab Pi Estimator ‘mypi.m’ % Graphics (no C++ equivalent) pivals = zeros(size(mypivals)); pivals = pi; mypimin = min(mypivals); mypimax = max(mypivals); v = [0 ntry mypimin mypimax]; axis(v); plot([1:ntry],mypivals,[1:ntry],pivals); xlabel('Number of random samples'); ylabel('Estimate of Pi'); hold off nsample = 100000; ntry = 500; ntotal = 0; mypivals = zeros(ntry,1); xy = zeros(nsample,2); ninside = 0; % number of (x,y) within the circle for j=1:ntry % random x and y between -1/2 and 1/2 xy=rand(nsample,2) - 0.5; for k=1:nsample x = xy(k,1); y = xy(k,2); % distance from center of circle r = sqrt(x*x+y*y); if r <= 0.5 ninside = ninside + 1; end end ntotal = ntotal + nsample; mypivals(j) = 4*ninside/ntotal; end mypi_estimate = 4*ninside/ntotal CSCI 317 Mike Heroux

  5. Estimate vs. True Pi CSCI 317 Mike Heroux

  6. Monte Carlo Methods • Methods that use random numbers this way are called Monte Carlo methods. • Other simple applications: • Estimating the likelihood of random events. • Example: Poker hands: What is the probability of 3-of-a-kind? • Modeling atomic states. • Approximate answers to otherwise intractable problems. CSCI 317 Mike Heroux

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