Lecture 3: January 22

Darts and the Poisson distribution

This applet is generated by Darts.java:

// Darts.java

import comphys.*;
import comphys.graphics.*;

public class Darts extends Animation {

    int L = 100;                // equal sized regions on dart board
    int N = 50;                 // number of darts
    int nTrial = 1000;          // number of trials of N throws each

    int[] B;                    // dart board
    int[] H;                    // histogram of hits in one trial
    int[] SumH;                 // sum of histograms
    double[] P;                 // probability distribution
    double nAverage;            // 
    double nSquaredAverage;     // 
    int darts = 0;              // darts thrown
    int trials = 0;             // trials completed

    void throwDart () {
        int i = (int) (L * Math.random());
        ++B[i];
        ++darts;
    }

    void histogram () {
        // compute H(n) for this trial
        for (int i = 0; i < L; i++) {
            int n = B[i];
            ++H[n];
        }
        // accumulate H(n) values
        double norm = 0;
        for (int n = 0; n <= N; ++n) {
            SumH[n] += H[n];
            norm += SumH[n];
        }
        // compute averages
        nAverage = nSquaredAverage = 0;
        for (int n = 0; n <= N; n++) {
            P[n] = SumH[n] / norm;
            nAverage += n * P[n];
            nSquaredAverage += n * n * P[n];
        }
    }

    void resetBoard () {
        for (int n = 0; n <= N; n++)
            H[n] = 0;
        for (int i = 0; i < L; i++)
            B[i] = 0;
        ++trials;
        darts = 0;
    }

    void initial () {
        B = new int[L];
        H = new int[N + 1];
        SumH = new int[N + 1];
        P = new double[N + 1];
        nAverage = nSquaredAverage = 0;
        darts = trials = 0;
    }

    class Board extends Plot {
        Board () {
            setSize(300, 300);
            setBackground("lightGray");
        }

        String[] colors = {"white", "green", "blue", "red",
                           "yellow", "cyan", "magenta", "black"};

        public void paint () {
            clear();
            int k = 1;
            while (k * k <= L)
                ++k;
            setWindow(0, k, 0, k);
            k = (int) Math.sqrt(L);
            for (int i = 0; i < L; i++) {
                int y = (int) (i / (double) k);    // row
                int x = i % k;                     // column
                int color = B[i];
                if (color >= colors.length)
                    color = colors.length - 1;
                setColor(colors[color]);
                boxArea(x + 0.05, x + 0.95, y + 0.05, y + 0.95);
            }
        }
    }

    class Graph extends Plot {
        Graph () {
            setSize(300, 300);
            setBackground("white");
        }

        public void paint () {
            clear();
            double yMax = 0.01;
            int nMax = 0;
            for (int n = 0; n <= N; n++) {
                if (P[n] > yMax)
                    yMax = P[n];
                if (P[n] > 0)
                    nMax = n;
            }
            setColor("black");
            drawAxes(0, nMax + 1, 0, 1.2 * yMax);
            setColor("cyan");
            for (int n = 0; n <= nMax; n++) {
                boxArea(n, n + 1, 0, P[n]);
            }
            setColor("blue");
            double x = 0.6 * (nMax + 1);
            double y = 1.1 * yMax;
            double dy = 0.1 * yMax;
            plotString("Trial = " + trials, x, y);
            y -= dy;
            plotString(" = " + Easy.format(nAverage, 4), x, y);
            y -= dy;
            plotString(" = " + Easy.format(nSquaredAverage, 4), x, y);
            y -= dy;
            double sigma = Math.sqrt(nSquaredAverage - nAverage * nAverage);
            plotString("Std Dev = " + Easy.format(sigma, 4), x, y);
            setColor("magenta");
            plotString("P(n) vs. n", 0, 1.25 * yMax); 

            // plot Poisson distribution
            setColor("red");
            double p0 = 0;
            for (int n = 0; n < nMax; n++) {
                double p = Math.pow(nAverage, n) * Math.exp(-nAverage);
                for (int i = 1; i <= n; i++)
                    p /= i;
                plotLine(n, p0, n, p);
                plotLine(n, p, n + 1, p);
                p0 = p;
            }
        }
    }

    Board board;
    Graph graph;
    Reader LReader, NReader, nTrialReader;

    boolean fast;
    class FastButton extends HotButton {
        public void action () {
            fast = !fast;
            if (fast)
                fastButton.setLabel("Slow");
            else
                fastButton.setLabel("Fast");
        }
    }
    FastButton fastButton;

    public void init () {
        initial();
        add(board = new Board());
        add(graph = new Graph());
        add(LReader = new Reader("Regions L =", L));
        add(NReader = new Reader("Darts N =", N));
        add(nTrialReader = new Reader("# Trials", nTrial));
        add(fastButton = new FastButton());
        fastButton.setLabel("Fast");
        addControlPanel();
    }

    public void step () {
        if (darts == N)
            resetBoard();
        if (trials == nTrial)
            stopAnimation();
        else {
            if (fast)
                while (darts < N - 1)
                    throwDart();
            throwDart();
            board.repaint();
            if (darts == N) {
                histogram();
                graph.repaint();
            }
        }
    }

    public void reset () {
        L = LReader.readInt();
        N = NReader.readInt();
        nTrial = nTrialReader.readInt();
        initial();
        board.repaint();
        graph.repaint();
    }

    public static void main (String[] args) {
        Darts darts = new Darts();
        darts.frame("Darts and the Poisson distribution", 630, 450);
    }
}

Another Poisson applet

The Poisson distribution is actually named for Siméon Denis Poisson, who is also known for Poisson's equation in electrostatics and Poisson brackets in classical mechanics.