Course Content

Advanced Probability Theory

1. Additional Statements From The Probability Theory

Course Overview Absolutely Continuous and Discrete Random Variables Cumulative Distribution Functions and Probability Density Functions Characteristics of Random Variables Random Vectors Useful Properties of the Gaussian Distribution Challenge: Detecting Outliers Using 3-Sigma Rule

2. The Limit Theorems of Probability Theory

Law of Large Numbers Law of Large Numbers for Bernoulli Process Challenge: Estimate Mean Value Using Law of Large Numbers Central Limit Theorem Challenge: Application of the CLT to Solving Real Problem

3. Estimation of Population Parameters

General population. Samples. Population parameters.Momentum estimation. Maximum Likelihood Estimation Challenge: Estimate Parameters of Chi-square Distribution Unbiased Estimation Challenge: Checking Bias of An Estimation Using Simulation Consistent Estimation Efficient Estimation Confidence Intervals for Population Parameters Challenge: Confidence Interval for Exponential Distribution Parameter

4. Testing of Statistical Hypotheses

What is Statistic Hypothesis? Type 1 and Type 2 Errors What is P-value?Comparing Means of Two Different Datasets Challenge: Using CLT to Compare Mean Values of Non-Gaussian Datasets Challenge: Resampling Approach to Compare Mean Values of the Datasets Testing the Hypothesis of Independence of Two Random Variables

Random Vectors

Random vectors represent multiple related random variables grouped together. They're commonly used in Data Science to model systems with interconnected random quantities, such as datasets where each vector corresponds to a data point.

The probability distribution of a random vector is described by its joint distribution function, which shows how all variables in the vector are distributed simultaneously.

To create a simple random vector with n dimensions:

Generate n independent random variables, each following its distribution function;
Use these variables as coordinates for the vector;
Apply the multiplication rule to determine the joint distribution: f = f1 * f2 * ... * fn.

Discrete random vectors

For discrete values, the joint distribution is still described using the PMF, where the function takes a combination of coordinate values and returns their probability.

For example, consider tossing two coins and recording the results in a vector. The PMF for this vector will look like:


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import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import binom

# Define parameters for the binomial distribution
n = 1  # Number of trials
p = 0.5  # Probability of success

# Generate values for the two dimensions
x = np.arange(0, n + 1)
y = np.arange(0, n + 1)
X, Y = np.meshgrid(x, y)

# Compute the PMF for each pair of values in the grid
pmf = binom.pmf(X, n, p) * binom.pmf(Y, n, p)
print(pmf)

Continuous random vectors

For continuous variables, multivariate PDF is used:


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import numpy as np
import matplotlib.pyplot as plt

# Define mean and covariance matrix for the Gaussian distribution
mu = np.array([0, 0])  # Mean vector
cov = np.array([[1, 0], [0, 1]])  # Covariance matrix

# Generate values for the two dimensions
x = np.linspace(-3, 3, 100)
y = np.linspace(-3, 3, 100)
X, Y = np.meshgrid(x, y)
pos = np.dstack((X, Y))

# Compute the PDF for each pair of values in the grid
pdf = (1 / (2 * np.pi * np.sqrt(np.linalg.det(cov)))) * np.exp(-0.5 * np.sum(np.dot(pos - mu, np.linalg.inv(cov)) * (pos - mu), axis=2))

# Plot the 2D PDF
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(X, Y, pdf, cmap='viridis')
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('PDF')
ax.set_title('Two-dimensional Gaussian PDF')
plt.show()

Note
The characteristics of random vectors are also given in vector form: this vector consists of coordinates that correspond to the statistics of the coordinates of the original vector. In the example above mu vector corresponds to the mean values of the coordinates, cov corresponds to the covariance matrix of a two-dimensional random vector.

Vectors with dependent coordinates

But there are also vectors with coordinates dependent on each other. Then, we will no longer be able to define the joint distribution as the product of the distributions of coordinates. In this case, the joint distribution is given based on the knowledge of the domain area or some additional information about the dependencies between the coordinates.

Let's look at the example of two-dimensional Gaussian samples with dependent coordinates. The nature of the dependencies will be determined using the covariance matrix (off-diagonal elements are responsible for the dependencies between the corresponding coordinates):


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import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import multivariate_normal

# Define mean vector and covariance matrix for the Gaussian distribution
mu = np.array([1, 3])  # Mean vector
cov = np.array([[4, -4], [-4, 5]])  # Covariance matrix

# Generate samples from the Gaussian distribution
samples = multivariate_normal.rvs(mean=mu, cov=cov, size=1000)

# Plot the sampled data
plt.scatter(samples[:, 0], y = samples[:, 1])
plt.xlabel('X')
plt.ylabel('Y')
plt.title('Samples from a Two-dimensional Gaussian Distribution with dependant coordinates')
plt.show()

We can see that these coordinates exhibit a strong linear dependency: as the X coordinate increases, the Y coordinate decreases.

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Section 1. Chapter 5