import numpy as np

import matplotlib.pyplot as plt

ϵ_values = np.random.randn(100)

rng = np.random.default_rng()

ϵ_values = rng.standard_normal(100)

plt.plot(ϵ_values)

plt.show()

```

```{index} single: Python; Subpackages

```

Consider the line `ϵ_values = np.random.randn(100)`.

Consider the line `rng = np.random.default_rng()`.

Here `np` refers to the package NumPy, while `random` is a **subpackage** of NumPy.

after the import statements are

```{code-cell} ipython

ϵ_values = np.random.randn(100)

ϵ_values = rng.standard_normal(100)

plt.plot(ϵ_values)

plt.show()

```

ϵ_values = [] # empty list

for i in range(ts_length):

e = np.random.randn()

e = rng.standard_normal()

ϵ_values.append(e)

plt.plot(ϵ_values)

```{code-cell} python3

for i in range(ts_length):

e = np.random.randn()

e = rng.standard_normal()

ϵ_values.append(e)

```

ϵ_values = []

i = 0

while i < ts_length:

e = np.random.randn()

e = rng.standard_normal()

ϵ_values.append(e)

i = i + 1

plt.plot(ϵ_values)

T = 200

x = np.empty(T+1)

x[0] = 0

rng = np.random.default_rng()

for t in range(T):

x[t+1] = α * x[t] + np.random.randn()

x[t+1] = α * x[t] + rng.standard_normal()

plt.plot(x)

plt.show()

α_values = [0.0, 0.8, 0.98]

T = 200

x = np.empty(T+1)

rng = np.random.default_rng()

for α in α_values:

x[0] = 0

for t in range(T):

x[t+1] = α * x[t] + np.random.randn()

x[t+1] = α * x[t] + rng.standard_normal()

plt.plot(x, label=f'$\\alpha = {α}$')

plt.legend()

T = 200

x = np.empty(T+1)

x[0] = 0

rng = np.random.default_rng()

for t in range(T):

x[t+1] = α * np.abs(x[t]) + np.random.randn()

x[t+1] = α * np.abs(x[t]) + rng.standard_normal()

plt.plot(x)

plt.show()

T = 200

x = np.empty(T+1)

x[0] = 0

rng = np.random.default_rng()

for t in range(T):

if x[t] < 0:

abs_x = - x[t]

else:

abs_x = x[t]

x[t+1] = α * abs_x + np.random.randn()

x[t+1] = α * abs_x + rng.standard_normal()

plt.plot(x)

plt.show()

T = 200

x = np.empty(T+1)

x[0] = 0

rng = np.random.default_rng()

for t in range(T):

abs_x = - x[t] if x[t] < 0 else x[t]

x[t+1] = α * abs_x + np.random.randn()

x[t+1] = α * abs_x + rng.standard_normal()

plt.plot(x)

plt.show()

```{code-cell} python3

n = 1000000 # sample size for Monte Carlo simulation

rng = np.random.default_rng()

count = 0

for i in range(n):

# drawing random positions on the square

u, v = np.random.uniform(), np.random.uniform()

u, v = rng.uniform(), rng.uniform()

# check whether the point falls within the boundary

# of the unit circle centred at (0.5,0.5)