#!/usr/bin/env python
# coding: utf-8
# 
# In[1]:
#Running this cell displays a button to toggle hidden code
#From: http://chris-said.io/2016/02/13/how-to-make-polished-jupyter-presentations-with-optional-code-visibility/
from IPython.display import HTML
HTML('''
''')
# # Central Limit Theorem Experiments
#
# **Please note: If you have hidden the code using the toggle in the cell above, the cells will appear as an empty line. However, they can still be run by pressing shift-enter with the empty cell selected.**
# ### Experiment 1(a):
#
# Toss a fair coin. If it lands heads, you win 100 dollars, and if it lands tails, you win 0 dollars. Run this experiment X times, where X is chosen by the user. We get a graph of the results, and discover that we get roughly a mean of 50 dollars and a standard deviation of 50 dollars.
#
# Note: In this plot and all others, the mean will be displayed as a vertical red line, and one standard deviation away from the mean on both sides will be displayed as a horizontal red line.
# In[2]:
#EXPERIMENT 1A
#SETUP
import numpy as np
import plotly
import plotly.offline as py
import plotly.graph_objs as go
py.init_notebook_mode(connected=False)
get_ipython().run_line_magic('matplotlib', 'inline')
#Takes input from the user for the value of X:
print("How many times would you like to run the experiment? ")
X = int(input())
X1 = X
#Stores the data on wins and losses:
win_data = []
#Performs the experiment X times:
for n in range(0, X):
#Flips a coin. 100 represents heads, 0 represents tails:
flip = np.random.choice((100, 0))
#Updates win_data with a win or a loss:
win_data.append(flip)
#Calculates the mean:
mean = np.mean(win_data)
#Calculates the standard deviation:
sd = np.std(win_data)
#Graphs a histogram of the data:
data = [go.Histogram(x = win_data, xbins=dict(
start=-10,
end=110,
size=10
))]
title = "Experiment 1(a) Winnings (Mean: {0:.2f}, Standard deviation: {1:.2f})".format(mean, sd)
layout = go.Layout(
title=title,
xaxis=dict(
title='Winnings',
range = [0,110],
fixedrange = True,
),
yaxis=dict(
title='Count',
fixedrange = False
),
shapes=[dict({
'type': 'line',
'x0': mean,
'y0': 0,
'x1': mean,
'y1': 1,
'xref': 'x',
'yref': 'paper',
'line': {
'color': '#f44242',
'width': 3
}}),
dict({
'type': 'line',
'x0': mean - sd,
'y0': 0,
'x1': mean + sd,
'y1': 0,
'line': {
'color': '#f44242',
'width': 10
}})]
)
fig = go.Figure(data=data, layout=layout)
py.iplot(fig, filename='Experiment 1 Data')
# ### Experiment 1(b)
#
# Toss a fair coin one hundred times, getting 1 dollar for each head and 0 dollars for each tail. Run this 100-trial experiment X times, where X is chosen by the user. We get a graph of the results, and discover that we get roughly a mean of 50 dollars and a standard deviation of 5 dollars.
# In[3]:
#EXPERIMENT 1B
#Takes input from the user for the value of X:
print("How many times would you like to run the experiment? ")
X = int(input())
X2 = X
#Stores the data on wins and losses:
win_data = []
#Performs the experiment X times:
for n in range(0, X1):
#Stores the amount won:
winnings = 0
#Flips 100 coins:
for m in range(0, 100):
#Flips a coin. 1 represents heads, 0 represents tails:
flip = np.random.choice((1, 0))
#Updates winnings:
winnings += flip
#Updates win_data with the final total for this run of the experiment:
win_data.append(winnings)
#Calculates the mean:
mean = np.mean(win_data)
#Calculates the standard deviation:
sd = np.std(win_data)
#Graphs a histogram of the data:
data = [go.Histogram(x = win_data, xbins=dict(
start=-10,
end=110,
size=5
))]
title = "Experiment 1(b) Winnings (Mean: {0:.2f}, Standard deviation: {1:.2f})".format(mean, sd)
layout = go.Layout(
title=title,
xaxis=dict(
title='Winnings',
range = [0,110],
fixedrange = True,
),
yaxis=dict(
title='Count',
fixedrange = False
),
shapes=[dict({
'type': 'line',
'x0': mean,
'y0': 0,
'x1': mean,
'y1': 1,
'xref': 'x',
'yref': 'paper',
'line': {
'color': '#f44242',
'width': 3
}}),
dict({
'type': 'line',
'x0': mean - sd,
'y0': 0,
'x1': mean + sd,
'y1': 0,
'line': {
'color': '#f44242',
'width': 10
}})]
)
fig = go.Figure(data=data, layout=layout)
py.iplot(fig, filename='Experiment 1 Data')
# ### Experiment 2(a)
#
# For experiments 2(a) and 2(b), we'll re-run the above experiments, and this time you can specify the value of N.
#
# Flip X coins (with the same X from experiment 1(a)), and on each coin flip, you win N dollars or win 0 dollars.
# In[4]:
#EXPERIMENT 2A
#Takes input from the user for the value of N:
print("How much money will a winning flip be worth? ")
N = int(input())
print(f"The experiment will be run {X1} times, as in experiment 1(a).")
#Stores the data on wins and losses:
win_data = []
#Performs the experiment X times:
for n in range(0, X):
#Flips a coin. N represents heads, 0 represents tails:
flip = np.random.choice((N, 0))
#Updates win_data with a win or a loss:
win_data.append(flip)
#Calculates the mean:
mean = np.mean(win_data)
#Calculates the standard deviation:
sd = np.std(win_data)
#Graphs a histogram of the data:
data = [go.Histogram(x = win_data, xbins=dict(
start=min(win_data),
end=max(win_data) + (max(win_data) - min(win_data))/15,
size=(max(win_data) - min(win_data))/15
))]
title = "Experiment 2(a) Winnings (Mean: {0:.2f}, Standard deviation: {1:.2f})".format(mean, sd)
layout = go.Layout(
title=title,
xaxis=dict(
title='Winnings',
#range = [0,110],
#fixedrange = True,
),
yaxis=dict(
title='Count',
fixedrange = False
),
shapes=[dict({
'type': 'line',
'x0': mean,
'y0': 0,
'x1': mean,
'y1': 1,
'xref': 'x',
'yref': 'paper',
'line': {
'color': '#f44242',
'width': 3
}}),
dict({
'type': 'line',
'x0': mean - sd,
'y0': 0,
'x1': mean + sd,
'y1': 0,
'line': {
'color': '#f44242',
'width': 10
}})]
)
fig = go.Figure(data=data, layout=layout)
py.iplot(fig, filename='Experiment 1 Data')
# ### Experiment 2(b)
#
# Toss a fair coin N times, getting 1 dollar for each head and 0 dollars for each tail. Run this N-trial experiment X times, where X is the same as specified in experiment 1(b).
# In[5]:
#EXPERIMENT 2B
print(f"The experiment will be run {X2} times, as in experiment 1(b).")
#Stores the data on wins and losses:
win_data = []
#Performs the experiment X times:
X = X2
for n in range(0, X):
#Stores the amount won:
winnings = 0
#Flips 100 coins:
for m in range(0, N):
#Flips a coin. 1 represents heads, 0 represents tails:
flip = np.random.choice((1, 0))
#Updates winnings:
winnings += flip
#Updates win_data with the final total for this run of the experiment:
win_data.append(winnings)
#Calculates the mean:
mean = np.mean(win_data)
#Calculates the standard deviation:
sd = np.std(win_data)
#Graphs a histogram of the data:
data = [go.Histogram(x = win_data, xbins=dict(
start=min(win_data),
end=max(win_data) + (max(win_data) - min(win_data))/15,
size=(max(win_data) - min(win_data))/15
))]
title = "Experiment 2(b) Winnings (Mean: {0:.2f}, Standard deviation: {1:.2f})".format(mean, sd)
layout = go.Layout(
title=title,
xaxis=dict(
title='Winnings',
#range = [0,110],
#fixedrange = True,
),
yaxis=dict(
title='Count',
fixedrange = False
),
shapes=[dict({
'type': 'line',
'x0': mean,
'y0': 0,
'x1': mean,
'y1': 1,
'xref': 'x',
'yref': 'paper',
'line': {
'color': '#f44242',
'width': 3
}}),
dict({
'type': 'line',
'x0': mean - sd,
'y0': 0,
'x1': mean + sd,
'y1': 0,
'line': {
'color': '#f44242',
'width': 10
}})]
)
fig = go.Figure(data=data, layout=layout)
py.iplot(fig, filename='Experiment 1 Data')
# ### Experiment 3(a)
# Toss a 6-sided die. In this experiment, we toss X of these dice, where X is specified by the user. We determine the mean and standard deviation of this experiment.
# In[6]:
#EXPERIMENT 3A
#Takes input from the user for the value of X:
print("How many 6-sided dice would you like to roll? ")
X = int(input())
X3 = X
#Stores the data:
win_data = []
#Performs the experiment X times:
for n in range(0, X):
#Rolls a die:
roll = np.random.choice((1,2,3,4,5,6))
#Updates win_data with the roll value:
win_data.append(roll)
#Calculates the mean:
mean = np.mean(win_data)
#Calculates the standard deviation:
sd = np.std(win_data)
#Graphs a histogram of the data:
data = [go.Histogram(x = win_data, xbins=dict(
start=min(win_data) - .5,
end=(max(win_data) + (max(win_data) - min(win_data))/15) + .5,
size=1
))]
title = "Experiment 3(a) Results (Mean: {0:.2f}, Standard deviation: {1:.2f})".format(mean, sd)
layout = go.Layout(
title=title,
xaxis=dict(
title='Value',
#range = [0,110],
#fixedrange = True,
),
yaxis=dict(
title='Count',
fixedrange = False
),
shapes=[dict({
'type': 'line',
'x0': mean,
'y0': 0,
'x1': mean,
'y1': 1,
'xref': 'x',
'yref': 'paper',
'line': {
'color': '#f44242',
'width': 3
}}),
dict({
'type': 'line',
'x0': mean - sd,
'y0': 0,
'x1': mean + sd,
'y1': 0,
'line': {
'color': '#f44242',
'width': 10
}})]
)
fig = go.Figure(data=data, layout=layout)
py.iplot(fig, filename='Experiment 1 Data')
# ### Experiment 3(b)
# Toss a 6-sided die. In this experiment, we toss 4 of these dice, and mark the average. We do this X times, where X is specified by the user. We determine the mean and standard deviation of this experiment, showing that the standard deviation reduces by a factor of sqrt(4).
# In[7]:
#EXPERIMENT 3B
#Takes input from the user for the value of X:
print("How many experiments would you like to run? ")
X = int(input())
X4 = X
#Stores the data:
mean_data = []
#Runs X experiments:
for m in range(0, X):
#Stores the results of the 4 rolls in this experiment
temp_data = []
#Rolls 4 dice:
for n in range(0, 4):
#Rolls a die:
roll = np.random.choice((1,2,3,4,5,6))
#Updates temp_data with the roll value:
temp_data.append(roll)
#Takes the average of the 4 rolls, appends it to mean_data:
mean = np.mean(temp_data)
mean_data.append(mean)
#Calculates the mean:
mean = np.mean(mean_data)
#Calculates the standard deviation:
sd = np.std(mean_data)
#Graphs a histogram of the data:
data = [go.Histogram(x = mean_data, xbins=dict(
start=min(mean_data),
end=max(mean_data) + (max(mean_data) - min(mean_data))/15,
size=(max(mean_data) - min(mean_data))/15
))]
title = "Experiment 3(b) Results (Mean: {0:.2f}, Standard deviation: {1:.2f})".format(mean, sd)
layout = go.Layout(
title=title,
xaxis=dict(
title='Value',
#range = [0,110],
#fixedrange = True,
),
yaxis=dict(
title='Count',
fixedrange = False
),
shapes=[dict({
'type': 'line',
'x0': mean,
'y0': 0,
'x1': mean,
'y1': 1,
'xref': 'x',
'yref': 'paper',
'line': {
'color': '#f44242',
'width': 3
}}),
dict({
'type': 'line',
'x0': mean - sd,
'y0': 0,
'x1': mean + sd,
'y1': 0,
'line': {
'color': '#f44242',
'width': 10
}})]
)
fig = go.Figure(data=data, layout=layout)
py.iplot(fig, filename='Experiment 1 Data')
# ### Experiment 4
# Re-run the above experiment, except we change 4 with N. What happens to the standard deviations as N gets larger?
# In[8]:
#EXPERIMENT 4
#Takes input from the user for the value of X:
print("How many 6-sided dice would you like to roll for each experiment? ")
N = int(input())
print(f"The experiment will be run {X4} times, as in experiment 3(b).")
X = X4
#Stores the data:
mean_data = []
#Runs X experiments:
for m in range(0, X):
#Stores the results of the N rolls in this experiment
temp_data = []
#Rolls N dice:
for n in range(0, N):
#Rolls a die:
roll = np.random.choice((1,2,3,4,5,6))
#Updates temp_data with the roll value:
temp_data.append(roll)
#Takes the average of the N rolls, appends it to mean_data:
mean = np.mean(temp_data)
mean_data.append(mean)
#Calculates the mean:
mean = np.mean(mean_data)
#Calculates the standard deviation:
sd = np.std(mean_data)
print()
print("The mean of the average rolls was: {:.3f}".format(mean))
print("The standard deviation was: {:.3f}".format(sd))
# ### Experiment 5
#
# Toss a 6-sided die, but this time, the six values are chosen by the user. (For example, it can be 1,2,3,4,5,10000). The user will specify X (the number of dice to be rolled for an individual trial) and N (the number of trials). The average of each of the N trials will be plotted.
#
# As a starting point, select the dice numbers to be 1,2,3,4,5,10, then select N=100 and X=10000. See how "normal" the result is.
#
# Experiment with different values to reproduce the Central Limit Theorem!
# In[23]:
#EXPERIMENT 5
#Choosing values for the die:
print("Choose what you want the values on the dice to be:")
print()
print("First value: ")
V1 = int(input())
print("Second value: ")
V2 = int(input())
print("Third value: ")
V3 = int(input())
print("Fourth value: ")
V4 = int(input())
print("Fifth value: ")
V5 = int(input())
print("Sixth value: ")
V6 = int(input())
values = [V1,V2,V3,V4,V5,V6]
print()
print("The values you have chosen are:")
print(values)
#Asking for the number of times the dice will be rolled per trial:
print()
print("Please choose a value for N - the number of dice we will be rolling for an individual trial.")
N = int(input())
#Asking for the number of trials:
print()
print("Please choose a value for X - the number of trials for which we will roll N dice.")
X = int(input())
sample_means = []
#Performing all trials:
for n in range(0, X):
sample = []
#Rolling all dice for a trial:
for m in range(0, N):
num = np.random.choice(values)
sample.append(num)
#Calculating mean of a trial:
mean = sum(sample)/len(sample)
sample_means.append(mean)
#Plotting the sample means:
data = [go.Histogram(x = sample_means)]
mean = sum(sample_means)/len(sample_means)
sd = np.std(sample_means)
title = "Sample Means (Mean: {0:.2f}, Standard deviation: {1:.2f})".format(mean, sd)
layout = go.Layout(
title=title,
xaxis=dict(
title='Value',
#range = [min(sample_means),max(sample_means)+1],
fixedrange = True,
#tick0 = 0,
#dtick = 1
),
yaxis=dict(
title='Count',
fixedrange = False
),
shapes=[dict({
'type': 'line',
'x0': mean,
'y0': 0,
'x1': mean,
'y1': 1,
'xref': 'x',
'yref': 'paper',
'line': {
'color': '#f44242',
'width': 3
}}),
dict({
'type': 'line',
'x0': mean - sd,
'y0': 0,
'x1': mean + sd,
'y1': 0,
'line': {
'color': '#f44242',
'width': 10
}})])
fig = go.Figure(data=data, layout=layout)
py.iplot(fig, filename='Sample')
# #### 