By Christopher van Hoecke, Max Margenot, and Delaney Mackenzie
https://www.quantopian.com/lectures/random-variables
This lecture corresponds to the Random Variables lecture, which is part of the Quantopian lecture series. This homework expects you to rely heavily on the code presented in the corresponding lecture. Please copy and paste regularly from that lecture when starting to work on the problems, as trying to do them from scratch will likely be too difficult.
Part of the Quantopian Lecture Series:
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# Useful Functions
class DiscreteRandomVariable:
def __init__(self, a=0, b=1):
self.variableType = ""
self.low = a
self.high = b
return
def draw(self, numberOfSamples):
samples = np.random.randint(self.low, self.high, numberOfSamples)
return samples
class BinomialRandomVariable(DiscreteRandomVariable):
def __init__(self, numberOfTrials = 10, probabilityOfSuccess = 0.5):
self.variableType = "Binomial"
self.numberOfTrials = numberOfTrials
self.probabilityOfSuccess = probabilityOfSuccess
return
def draw(self, numberOfSamples):
samples = np.random.binomial(self.numberOfTrials, self.probabilityOfSuccess, numberOfSamples)
return samples
def factorial(n):return reduce(lambda x,y:x*y,[1]+range(1,n+1))
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# Useful Libraries
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import statsmodels.stats as stats
from statsmodels.stats import stattools
from __future__ import division
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# Histograms with 10 tosses.
## Your code goes here
plt.xlabel('Value')
plt.ylabel('Occurences')
plt.legend(['Coin Tosses']);
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# Histograms with 1000000 tosses.
## Your code goes here
plt.xlabel('Value')
plt.ylabel('Occurences')
plt.legend(['Coin Tosses']);
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# Binomial distribution with p=0.25 and n=20
## Your code goes here.
plt.title('Binomial Distributino with p=0.25 and n=20')
plt.xlabel('Value')
plt.ylabel('Occurences')
plt.legend(['Die Rolls']);
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# Finding x which occurs most often
## Your code goes here
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# Calculating the probability of finding x.
## Your code goes here
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# Graphin a normal distribution pdf.
## Your code goes here
mu =
sigma =
x = np.linspace(-30, 30, 200)
y =
plt.plot(x, y)
plt.title('Graph of PDF with mu = 0 and sigma = 5')
plt.xlabel('Value')
plt.ylabel('Probability');
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# finding the 1st, 2nd, and third confidence intervals.
## Your code goes here
first_ci =
second_ci =
third_ci =
print '1-sigma -> mu +/-', sigma
print '2-sigma -> mu +/-', second_ci[1]
print '3-sigma -> mu +/-', third_ci[1]
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## Graphing.
## Your code goes here.
plt.title('Graph of PDF with 3 confidence intervals.')
plt.legend();
Fit the returns of SPY from 2016-01-01 to 2016-05-01 to a normal distribution.
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# Collect prices and retursn.
prices = get_pricing('SPY', start_date = '2016-01-01', end_date='2016-05-01',
fields = 'price')
returns = prices.pct_change()[1:]
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# Calculating the mean and standard deviation.
## Your code goes here
sample_mean =
sample_std_dev =
x = np.linspace(-(sample_mean + 4 * sample_std_dev), (sample_mean + 4 * sample_std_dev), len(returns))
sample_distribution = ((1/(sample_std_dev * 2 * np.pi)) *
np.exp(-(x - sample_mean)*(x - sample_mean) / (2 * sample_std_dev * sample_std_dev)))
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# Plotting histograms and confidence intervals.
## Your code goes here
plt.title('Graph of returns with fitted PDF and the 3 confidence intervals. ')
plt.legend();
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# Run the JB test for normality.
## Your code goes here
print "The JB test p-value is: ", p_value
print "We reject the hypothesis that the data are normally distributed ", p_value < cutoff
print "The skewness of the returns is: ", skewness
print "The kurtosis of the returns is: ", kurtosis
Congratulations on completing the Random Variables exercises!
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