Challenge: Estimate Mean Value Using Law of Large Numbers

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Assume that we have some data samples: we know these samples are independent and identically distributed, but we do not know the characteristics.

Your task is to use the law of large numbers to estimate the expected value of these samples.
We will also try to check the assumption that our data has exponential distribution: we will build a histogram based on our data and compare it with the real PDF of the exponential distribution.

Note

Visualization cannot prove that the data is distributed in a certain way. For this, it is necessary to use statistical tests, which will be considered in the last section of this course; however, with the help of visualization, we can at least roughly determine which class of distributions our data belongs to.

Your task is to:

Plot histogram using .hist() method of matplotlib.pyplot module.
Calculate the mean over a given subsample in mean_value function using .mean() method.
Pass exp_samples as an argument of a function to calculate mean values over all subsamples.
Print the estimated mean value of all samples as the last value of y array.

Solución

import pandas as pd

import matplotlib.pyplot as plt

import numpy as np

exp_samples = pd.read_csv('https://codefinity-content-media.s3.eu-west-1.amazonaws.com/Advanced+Probability+course+media/expon_samples.csv', names = ['Value'])

# Plot a histogram of the samples and pdf of expon distribution

fig, axes = plt.subplots(1,2)

fig.set_size_inches(10, 5)

axes[0].hist(exp_samples.values, bins=10, alpha=0.5, edgecolor='black', density=True)

axes[0].set_title('Histogram of Data Samples')

# Define the range of x values to plot

x = np.linspace(0, 10, 1000)

# Calculate the PDF of the exponential distribution at each x value

pdf = 0.5 * np.exp(-0.5 * x)

axes[1].plot(x, pdf)

axes[1].set_title('Exponential Distribution PDF')

plt.show()

# Function that will calculate mean value of subsamples

def mean_value(data, subsample_size):

return data[:subsample_size].mean()['Value']

# Visualizing the results

x = np.arange(5000)

y = np.zeros(5000)

for i in range(1, 5000):

y[i] = mean_value(exp_samples, x[i])

plt.plot(x, y, label='Estimated mean')

plt.xlabel('Number of elements to calculate mean value')

plt.ylabel('Mean value')

plt.title('Mean value of the samples')

plt.legend()

plt.show()

¿Todo estuvo claro?

¡Gracias por tus comentarios!

Sección 2. Capítulo 3

import pandas as pd

import matplotlib.pyplot as plt

import numpy as np

exp_samples = pd.read_csv('https://codefinity-content-media.s3.eu-west-1.amazonaws.com/Advanced+Probability+course+media/expon_samples.csv', names = ['Value'])

# Plot a histogram of the samples and pdf of expon distribution

fig, axes = plt.subplots(1,2)

fig.set_size_inches(10, 5)

axes[0].___(exp_samples.values, bins=10, alpha=0.5, edgecolor='black', density=True)

axes[0].set_title('Histogram of Data Samples')

# Define the range of x values to plot

x = np.linspace(0, 10, 1000)

# Calculate the PDF of the exponential distribution at each x value

pdf = 0.5 * np.exp(-0.5 * x)

axes[1].plot(x, pdf)

axes[1].set_title('Exponential Distribution PDF')

plt.show()

# Function that will calculate mean value of subsamples

def mean_value(data, subsample_size):

return data[:subsample_size].___()['Value']

# Visualizing the results

x = np.arange(5000)

y = np.zeros(5000)

for i in range(1, 5000):

y[i] = mean_value(___, x[i])

plt.plot(x, y, label='Estimated mean')

plt.xlabel('Number of elements to calculate mean value')

plt.ylabel('Mean value')

plt.title('Mean value of the samples')

plt.legend()

plt.show()

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