Hladina v nádržích¶

In [1]:

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import axes3d
import sympy as sp

In [2]:

%matplotlib notebook
sp.init_printing()
x, y = sp.symbols('x y')

Řešení¶

sdružená hustota pravděpodobnosti¶

$$f_{XY}(x,y) = \frac{6}{5} \left( x + y^2 \right)$$

In [3]:

fxy = sp.Rational(6, 5) * (x + y**2)
display(fxy)
print('Plocha fxy =', sp.integrate(sp.integrate(fxy, (x, 0, 1)), (y, 0, 1)))

$$\frac{6 x}{5} + \frac{6 y^{2}}{5}$$

Plocha fxy = 1

distribuční funkce¶

In [4]:

Fxy = sp.integrate(sp.integrate(fxy, (x, 0, x)), (y, 0, y))
display(Fxy)

$$\frac{3 y}{5} x^{2} + \frac{2 x}{5} y^{3}$$

marginální hustoty pravděpodobnosti¶

In [5]:

fx = sp.integrate(fxy, (y, 0, 1))
display(fx)
print('Plocha fx =', sp.integrate(fx, (x, 0, 1)))

fy = sp.integrate(fxy, (x, 0, 1))
display(fy)
print('Plocha fy =', sp.integrate(fy, (y, 0, 1)))

$$\frac{6 x}{5} + \frac{2}{5}$$

Plocha fx = 1

$$\frac{6 y^{2}}{5} + \frac{3}{5}$$

Plocha fy = 1

nezávislost $X$ a $Y$¶

In [6]:

print('Jsou X a Y nezávislé?', fx * fy == fxy)
fxy_mult = fx * fy
display(sp.simplify(fxy_mult))

Jsou X a Y nezávislé? False

$$\frac{6}{25} \left(3 x + 1\right) \left(2 y^{2} + 1\right)$$

pravděpodobnosti¶

In [7]:

print('P(X=0.5 | Y<0.5) =', sp.integrate(fxy.subs(dict(x=0.5)), (y, 0, .5)))
print('P(X<0.5 ^ Y<0.5) =', sp.integrate(sp.integrate(fxy, (x, 0, .5)), (y, 0, .5)))

P(X=0.5 | Y<0.5) = 0.350000000000000
P(X<0.5 ^ Y<0.5) = 0.100000000000000

Grafické zobrazení hustot¶

Zadaná sdružená hustota pravděpodobnosti¶

In [8]:

fig = plt.figure(tight_layout=True)
ax = fig.gca(projection='3d')

x = np.linspace(0, 1, 100)
y = np.linspace(0, 1, 100)
X, Y = np.meshgrid(x, y)
Z = 6 / 5 * (X + Y**2)
ax.plot_surface(X, Y, Z, rstride=8, cstride=8, cmap=plt.get_cmap('jet'), alpha=.5)
#ax.plot_wireframe(X, Y, Z, rstride=8, cstride=8, cmap=plt.get_cmap('coolwarm'), alpha=.)

cset = ax.contour(X, Y, Z, zdir='z', offset=0, cmap=plt.get_cmap('jet'))

ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z');

No description has been provided for this image

Sdružená hustota pravděpodobnosti získaná jako součin marginálních hustot původní sdružené hustoty pravděpodobnosti¶

In [9]:

fig = plt.figure(tight_layout=True)
ax = fig.gca(projection='3d')

x = np.linspace(0, 1, 100)
y = np.linspace(0, 1, 100)
X, Y = np.meshgrid(x, y)
Z = 6 * (3 * X + 1) * (2 * Y**2 + 1) / 25
ax.plot_surface(X, Y, Z, rstride=8, cstride=8, cmap=plt.get_cmap('jet'), alpha=.5)

cset = ax.contour(X, Y, Z, zdir='z', offset=0, cmap=plt.get_cmap('jet'))

ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z');

Marginální hustoty pravděpodobnosti¶

In [10]:

fig, ax = plt.subplots(tight_layout=True)
# fx
x = np.linspace(0, 1, 100)
y = 6*x/5 + 2/5
ax.plot(x, y, label='fx')

# fy
x = np.linspace(0, 1, 100)
y = 6*x**2/5 + 3/5
ax.plot(x, y, label='fy')

ax.legend();

Distribuční funkce původní sdružené hustoty pravděpodobnosti¶

In [12]:

fig = plt.figure(tight_layout=True)
ax = fig.gca(projection='3d')

x = np.linspace(0, 1, 100)
y = np.linspace(0, 1, 100)
X, Y = np.meshgrid(x, y)
Z = 3*X**2*Y/5 + 2*X*Y**3/5
ax.plot_surface(X, Y, Z, rstride=8, cstride=8, cmap=plt.get_cmap('jet'), alpha=.5)

#cset = ax.contour(X, Y, Z, 10, zdir='z', offset=1, cmap=plt.get_cmap('jet'))

ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z');

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nadrze