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)

1. 定数 c を求める。

$\begin{array}{}{\int }_{-\pi }^{\pi }f\left(x\right)\mathrm{dx}\\ =c{\int }_{0}^{\pi }\left(1-\mathrm{cos}x\right)\mathrm{dx}\\ =c{\left[x-\mathrm{sin}x\right]}_{0}^{\pi }\\ =\pi c\\ \pi c=1\\ c=\frac{1}{\pi }\end{array}$

f に付随する確率密度関数。

$\begin{array}{}-\pi \le x\le 0\\ F\left(x\right)\\ ={\int }_{-\pi }^{x}f\left(t\right)\mathrm{dt}\\ =0\\ 0\le x\le \pi \\ F\left(x\right)\\ ={\int }_{-\pi }^{x}f\left(t\right)\mathrm{dt}\\ ={\int }_{0}^{x}f\left(t\right)\mathrm{dt}\\ =\frac{1}{\pi }{\left[t-\mathrm{sin}t\right]}_{0}^{x}\\ =\frac{1}{\pi }\left(x-\mathrm{sin}x\right)\end{array}$

求める確率は、

$\begin{array}{}\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{1}{\pi }\left(1-\mathrm{cos}t\right)\mathrm{dt}\\ =\frac{1}{\pi }{\left[x-\mathrm{sin}x\right]}_{0}^{\frac{\pi }{2}}\\ =\frac{1}{\pi }\left(\frac{\pi }{2}-1\right)\\ =\frac{1}{2}-\frac{1}{\pi }\end{array}$

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')


$./sample12.py 12. π 0 ⌠ ⌠ ⎮ c⋅(-cos(x) + 1) dx - 1 + ⎮ 0 dx ⌡ ⌡ 0 -π ⎡1⎤ ⎢─⎥ ⎣π⎦ π ─ 2 0 ⌠ ⌠ ⎮ -cos(x) + 1 ⎮ 0 dx + ⎮ ─────────── dx ⌡ ⎮ π -π ⌡ 0 -2 + π ────── 2⋅π$


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,
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])
let yscale = d3.scaleLinear()
.domain([y1, y2])

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();