Probability Theory Mastering

1. Additional Statements From The Probability Theory

Now we will understand some fundamental theoretical concepts which are used in solving real live tasks: absolutely continuous and discrete random variables, probability density function, cumulative distribution function, the characteristics of a random variable, etc.

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

The limit theorems of probability theory are fundamental laws of probability theory that are often used in practice in a wide variety of areas, such as: building confidence intervals, estimating distribution parameters, providing A/B testings, creating ensembles of ML models, etc. Now we will consider two of the most commonly used: the Law of Large Numbers and the Central Limit Theorem.

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

When we work with real data we usually do not know from which distribution this data was obtained. In order to determine this, we must be able to correctly estimate the parameters of this distribution and the type of distribution, which we will learn to do in this section.

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

We have already learned how to estimate the parameters of the population. But to estimate the parameter, we make an assumption about the population distribution. Can we say that our assumption is correct? How do we prove that the estimated parameters are the real parameters of the population? Can we show that two sets of samples are independent? To answer these questions, it is necessary to consider the concept of hypothesis testing.

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

Challenge: Estimate Mean Value Using Law of Large Numbers

Task

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.

Everything was clear?

Section 2. Chapter 3

Course Content

Probability Theory Mastering

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

Probability Theory Mastering

Challenge: Estimate Mean Value Using Law of Large Numbers

Task

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

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.

Everything was clear?

Section 2. Chapter 3