数学 - Python - JavaScript - 解析学 - 積分 - 積分の応用 - 確率(区間、三角関数(正弦と余弦)、累乗(べき乗、平方)、確率密度関数、付随する確率関数)

学習環境

解析入門原書第3版 (S.ラング(著)、松坂和夫(翻訳)、片山孝次(翻訳)、岩波書店)の第3部(積分)、第13章(積分の応用)、6(確率)の練習問題12の解答を求めてみる。

定数 c を求める。

$\begin{array}{l} \int_{- π}^{π} f (x) dx \\ = c \int_{0}^{π} (1 - \cos x) dx \\ = c {[x - \sin x]}_{0}^{π} \\ = π c \\ π c = 1 \\ c = \frac{1}{π} \end{array}$

f に付随する確率密度関数。

$\begin{array}{l} - π \leq x \leq 0 \\ F (x) \\ = \int_{- π}^{x} f (t) dt \\ = 0 \\ 0 \leq x \leq π \\ F (x) \\ = \int_{- π}^{x} f (t) dt \\ = \int_{0}^{x} f (t) dt \\ = \frac{1}{π} {[t - \sin t]}_{0}^{x} \\ = \frac{1}{π} (x - \sin x) \end{array}$

求める確率は、

$\begin{array}{l} \int_{0}^{\frac{π}{2}} \frac{1}{π} (1 - \cos t) dt \\ = \frac{1}{π} {[x - \sin x]}_{0}^{\frac{π}{2}} \\ = \frac{1}{π} (\frac{π}{2} - 1) \\ = \frac{1}{2} - \frac{1}{π} \end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, solve, cos, pi, plot

print('12.')

c, x = symbols('c, x')

f = c * (1 - cos(x))

eq = Integral(0, (x, -pi, 0)) + Integral(f, (x, 0, pi)) - 1
cs = solve(eq.doit(), c)

for t in [eq, cs]:
    pprint(t)
    print()

fc = f.subs({c: cs[0]})

I = Integral(0, (x, -pi, 0)) + Integral(fc, (x, 0, pi / 2))

for t in [I, I.doit()]:
    pprint(t.simplify())
    print()

p = plot((0, (x, -pi, 0)),
         (fc, (x, 0, pi / 2)),
         (fc, (x, pi / 2, pi)),
         legend=True, show=False)
colors = ['red', 'green', 'red']
for i, color in enumerate(colors):
    p[i].line_color = color
p.save('sample12.png')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample12.py
12.
π                          0      
⌠                          ⌠      
⎮ c⋅(-cos(x) + 1) dx - 1 + ⎮  0 dx
⌡                          ⌡      
0                          -π     

⎡1⎤
⎢─⎥
⎣π⎦

          π               
          ─               
          2               
0         ⌠               
⌠         ⎮ -cos(x) + 1   
⎮  0 dx + ⎮ ─────────── dx
⌡         ⎮      π        
-π        ⌡               
          0               

-2 + π
──────
 2⋅π  

$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
 <input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">

<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="sample12.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 fns = [[x => x <= 0 ? 0 : 1 / Math.PI * (1 - Math.cos(x)), 'red']];

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 = [[-Math.PI, y1, -Math.PI, y2, 'green'],
                 [0, y1, 0, y2, 'blue'],
                 [Math.PI / 2, y1, Math.PI / 2, y2, 'brown'],
                 [Math.PI, y1, Math.PI, y2, 'orange']];

    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();

r = dx =
x1 = x2 =
y1 = y2 =

Kamimura's blog

2019年1月8日火曜日

数学 - Python - JavaScript - 解析学 - 積分 - 積分の応用 - 確率(区間、三角関数(正弦と余弦)、累乗(べき乗、平方)、確率密度関数、付随する確率関数)

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