2018年6月14日木曜日

学習環境

数学読本〈6〉線形写像・1次変換/数論へのプレリュード/集合論へのプレリュード/εとδ/落ち穂拾い など(松坂 和夫(著)、岩波書店)の第26章(エピローグ - 落ち穂拾い、など)、26.4(確率分布と平均)、確率変数の平均または期待値、問6-(1).を取り組んでみる。



    1. E X = n = 0 36 n · 36 n · 1 6 n 5 6 36 - n 1 6 x + 5 6 36 = n = 0 36 36 n 1 6 x n 5 6 36 - n

      両辺を x について微分する。

      36 · 1 6 1 6 x + 5 6 35 = n = 0 36 36 n 1 6 n 5 6 36 - n n x n - 1

      x に1を代入。

      6 = n = 0 36 36 n 1 6 n 5 6 36 - n n

      よって、求める平均は、

      E X = 6

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, combinatorial, Rational
import random

nC = combinatorial.numbers.nC
pprint(sum([n * nC(36, n) * Rational(1, 6) ** n * Rational(5, 6) ** (36 - n)
            for n in range(37)]))

n = 10
dice = list(range(1, 7))
for i in range(1, 6):
    n = 10 ** i
    t = 0
    for _ in range(n):
        nums = [random.choice(dice) for _ in range(36)]
        l = len(list(filter(lambda x: x == 1, nums)))
        t += l
    print(f'試行回数 {n}回、{t / n}')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample6.py
6
試行回数 10回、6.0
試行回数 100回、5.86
試行回数 1000回、6.007
試行回数 10000回、5.9752
試行回数 100000回、5.99674
$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="1">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.001" value="1">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="1">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="1000">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="36">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample6.js"></script>

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n';

let f = (x) => {
    let t = 0;
    for (let i = 0; i < x; i += 1) {
        let a = [];
        for (let i = 0; i < 36; i += 1) {
            a.push(1 + Math.floor(Math.random() * 6));
        }
        t  += a.filter((n) => n === 1).length;
    }
    return t / x;
};

let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        lines = [[x1, 6, x2, 6, 'red']],
        fns = [[f, 'green']];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                points.push([x, y, color]);
            }
        });
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);
    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', (d) => d[4] || 'black');

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d) => d[2] || 'green');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);

    [fns].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();







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