Python version of RunningProportion.R
In [1]:
%matplotlib inline
import matplotlib.pyplot as plt
plt.style.use('ggplot')
import numpy as np
np.random.seed(47405)
In [2]:
N = 500 # Specify the total number of flips, denoted N.
p_heads = 0.5 # Specify underlying probability of heads.
In [3]:
# Flip a coin N times and compute the running proportion of heads at each flip.
# Generate a random sample of N flips (heads=1, tails=0):
flip_sequence = np.random.choice([0,1], p=[1-p_heads,p_heads], size=N)
In [4]:
# Compute the running proportion of heads:
r = np.cumsum(flip_sequence) # Cumulative sum: Number of heads at each step.
n = np.linspace(1, N, num=N) # Number of flips at each step.
run_prop = r / n # Component by component division.
In [5]:
# Display the beginning of the flip sequence:
np.array(['T', 'H'])[flip_sequence[:10]].tolist()
Out[5]:
In [6]:
# Display the relative frequency at the end of the sequence.
run_prop[-1]
Out[6]:
In [7]:
# Graph the running proportion:
plt.figure(figsize=(10,6))
plt.plot(n, run_prop, marker='.')
plt.hlines(p_heads, n[0], n[-1], linestyle='dashed')
plt.ylim([0.0, 1.0])
plt.xscale('log')
plt.xlim(n[0], n[-1])
plt.title('Running Proportion of Heads')
plt.xlabel('Flip Number')
plt.ylabel('Proportion Heads')
plt.show()
In [8]:
import pandas as pd
pd.options.display.float_format = '{:.2f}'.format #
In [9]:
# load and preview data
df = pd.read_csv('../datasets/HairEyeColor.csv')
df.head()
Out[9]:
In [10]:
eye_hair_freq = df.groupby(['Eye', 'Hair']).sum() # Sum across sex
# p(eye, hair)
eye_hair_prop = eye_hair_freq / eye_hair_freq.sum() # joint proportions, Table 4.1
eye_hair_prop.rename(columns={'Freq': 'Prop'}, inplace=True)
eye_hair_prop
Out[10]:
In [11]:
hair_freq = df.groupby('Hair').sum() # Sum across sex and eye
# p(hair)
hair_prop = hair_freq / hair_freq.sum() # marginal proportions, Table 4.1
hair_prop.rename(columns={'Freq': 'Prop'}, inplace=True)
hair_prop
Out[11]:
In [12]:
eye_freq = df.groupby('Eye').sum() # Sum across sex and hair
# p(eye)
eye_prop = eye_freq / eye_freq.sum() # marginal proportions, Table 4.1
eye_prop.rename(columns={'Freq': 'Prop'}, inplace=True)
eye_prop
Out[12]:
In [13]:
# probabilities of the hair colors given Blue eyes, p(hair|Blue eyes)
eye_hair_prop.loc['Blue'] / eye_prop.loc['Blue'] # conditional prob, Table 4.2
Out[13]:
In [14]:
# probabilities of the hair colors given Brown eyes, p(hair|brown eyes)
eye_hair_prop.loc['Brown'] / eye_prop.loc['Brown']
Out[14]:
In [15]:
# probabilities of the eye colors given Brown hair, p(eyes|brown hair)
brown_hair_eye_prop = eye_hair_prop.xs('Brown', level=1)
brown_hair_eye_prop / hair_prop.loc['Brown']
Out[15]:
Set p_heads = 0.8 in 4.5. Appendix: Code for Figure 4.1 and re-run the section.
In [16]:
values = ['9', '10', 'J', 'Q', 'K', 'A']
suits = ['♠︎', '♣︎', '♥︎', '♦︎']
n_suits = len(suits)
n_values = len(values)
n_cards = 48
In [17]:
n_10 = 2 * n_suits
p_10 = n_10 / n_cards
p_10
Out[17]:
In [18]:
n_J = 2 * n_suits
p_10_or_J = (n_10 + n_J) / n_cards
p_10_or_J
Out[18]:
In [19]:
x, dx = np.linspace(0, 1, 50, retstep=True)
In [20]:
p = lambda x: 6 * x * (1 - x) # p(x)
y = p(x)
In [21]:
plt.plot(x, y)
plt.vlines(x, 0, y)
plt.xlabel('x')
plt.ylabel('p(x)')
plt.title('Probability Density, p(x)')
plt.show()
In [22]:
area = sum(y * dx)
area
Out[22]:
Part B
p(x) = 6*x*(1 - x) = 6*x - 6*x^2
p(x) = f'(x)
f(x) = 6*x^2/2 - 6*x^3/3 = 3x^2 - 2x^3
f(1) - f(0) = 3 - 2 = 1Part C
f(x) = 0; x - ?
3 * x^2 - 2 * x^3 = 0
3 - 2 * x = 0
x = 3/2 = 1.5Part A
In [23]:
meanval = 0.0 # Specify mean of distribution.
sdval = 0.2 # Specify standard deviation of distribution.
xlow = meanval - sdval # Specify low end of x-axis.
xhigh = meanval + sdval # Specify high end of x-axis.
In [24]:
x, dx = np.linspace(xlow, xhigh, 50, retstep=True)
In [25]:
p = lambda x: ( 1/(sdval*np.sqrt(2*np.pi)) ) * np.exp( -((x-meanval)/sdval)**2/2 )
y = p(x)
In [26]:
plt.plot(x, y)
plt.vlines(x, 0, y)
plt.xlim([x[0], x[-1]])
plt.xlabel('x')
plt.ylabel('p(x)')
plt.title('Probability Density, p(x)')
plt.show()
In [27]:
area = sum(y * dx)
area
Out[27]:
Part B
In [28]:
meanval = 162
sdval = (177 - 147) / 2
sdval
Out[28]:
In [29]:
# p(food|grade)
p_food_grade = [
[0.3, 0.6, 0.1],
[0.6, 0.3, 0.1],
[0.3, 0.1, 0.6]
]
p_food_grade = pd.DataFrame(p_food_grade,
index=['grade 1st','grade 6th', 'grade 11th'],
columns=['Ice cream', 'Fruit', 'French fries'])
p_food_grade
Out[29]:
In [30]:
# p(grade)
p_grade = pd.Series([0.2, 0.2, 0.6], index=['grade 1st','grade 6th', 'grade 11th'])
p_grade
Out[30]:
In [31]:
# p(grade, food)
p_join = p_food_grade.copy()
p_join['Ice cream'] = p_food_grade['Ice cream'] * p_grade
p_join['Fruit'] = p_food_grade['Fruit'] * p_grade
p_join['French fries'] = p_food_grade['French fries'] * p_grade
p_join
Out[31]:
In [32]:
p_join.values.sum()
Out[32]: