import numpy as np
import matplotlib.pyplot as plt

ϵ_values = np.random.randn(100)
plt.plot(ϵ_values)
plt.show()

np.sqrt(4)

np.log(4)

import numpy as np

np.sqrt(4)

from numpy import sqrt

sqrt(4)

ϵ_values = np.random.randn(100)
plt.plot(ϵ_values)
plt.show()

ts_length = 100
ϵ_values = []   # empty list

for i in range(ts_length):
    e = np.random.randn()
    ϵ_values.append(e)

plt.plot(ϵ_values)
plt.show()

x = [10, 'foo', False]
type(x)

x

x.append(2.5)
x

x

x.pop()

x

x[0]   # first element of x

x[1]   # second element of x

for i in range(ts_length):
    e = np.random.randn()
    ϵ_values.append(e)

animals = ['dog', 'cat', 'bird']
for animal in animals:
    print("The plural of " + animal + " is " + animal + "s")

ts_length = 100
ϵ_values = []
i = 0
while i < ts_length:
    e = np.random.randn()
    ϵ_values.append(e)
    i = i + 1
plt.plot(ϵ_values)
plt.show()

i == ts_length #the ending condition for the while loop

r = 0.025         # interest rate
T = 50            # end date
b = np.empty(T+1) # an empty NumPy array, to store all b_t
b[0] = 10         # initial balance

for t in range(T):
    b[t+1] = (1 + r) * b[t]

plt.plot(b, label='bank balance')
plt.legend()
plt.show()

import numpy as np
import matplotlib.pyplot as plt

α = 0.9
T = 200
x = np.empty(T+1)
x[0] = 0

for t in range(T):
    x[t+1] = α * x[t] + np.random.randn()

plt.plot(x)
plt.show()

α_values = [0.0, 0.8, 0.98]
T = 200
x = np.empty(T+1)

for α in α_values:
    x[0] = 0
    for t in range(T):
        x[t+1] = α * x[t] + np.random.randn()
    plt.plot(x, label=f'$\\alpha = {α}$')

plt.legend()
plt.show()

α = 0.9
T = 200
x = np.empty(T+1)
x[0] = 0

for t in range(T):
    x[t+1] = α * np.abs(x[t]) + np.random.randn()

plt.plot(x)
plt.show()

numbers = [-9, 2.3, -11, 0]

for x in numbers:
    if x < 0:
        print(-1)
    else:
        print(1)

α = 0.9
T = 200
x = np.empty(T+1)
x[0] = 0

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()

plt.plot(x)
plt.show()

α = 0.9
T = 200
x = np.empty(T+1)
x[0] = 0

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()

plt.plot(x)
plt.show()

import numpy as np

n = 1000000 # sample size for Monte Carlo simulation

count = 0
for i in range(n):

    # drawing random positions on the square
    u, v = np.random.uniform(), np.random.uniform()

    # check whether the point falls within the boundary
    # of the unit circle centred at (0.5,0.5)
    d = np.sqrt((u - 0.5)**2 + (v - 0.5)**2)

    # if it falls within the inscribed circle, 
    # add it to the count
    if d < 0.5:
        count += 1

area_estimate = count / n

print(area_estimate * 4)  # dividing by radius**2