#2022 Q2(4) 2024.8.1

import numpy as np
import scipy.integrate as integrate

# 積分の被積分関数を定義
def integrand(v, u):
    return 1/4 if (u**2 - 2*u*v + v**2 <= 1) else 0

# 積分の範囲を定義する関数
def bounds_u():
    return -1, 1

def bounds_v(u):
    return -1, 1

# 確率 P(U^2 - 2UV + V^2 <= 1) を計算
probability, _ = integrate.nquad(integrand, [bounds_v, bounds_u])

probability

0.7500047152195971

#2022 Q2(4) 2024.8.1

import matplotlib.pyplot as plt
import numpy as np

# UとVの範囲を設定
u = np.linspace(-1, 1, 400)
v = np.linspace(-1, 1, 400)
U, V = np.meshgrid(u, v)

# 条件 U^2 - 2UV + V^2 <= 1 を満たす領域を計算
condition = (U**2 - 2*U*V + V**2 <= 1)

# グラフの描画
plt.figure(figsize=(5, 5))
plt.contourf(U, V, condition, levels=[0.5, 1], colors=['lightblue'], alpha=0.5)
plt.title(r'条件 $U^2 - 2UV + V^2 \leq 1$')
plt.xlabel('U')
plt.ylabel('V')
plt.axhline(0, color='gray', lw=0.5)
plt.axvline(0, color='gray', lw=0.5)
plt.grid(True)

# 条件を満たす境界を描画
plt.contour(U, V, condition, levels=[0.5], colors=['blue'])

plt.show()