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: Application of the CLT to Solving Real Problem

Let's imagine that we need to solve the following problem:

Suppose that we come to the shooting range and start shooting, the probability of hitting the target is 0.4, respectively the probability of missing is 0.6;
We shot 100 times and needed to calculate the probability that the hits would be between 50 and 70.

We have a standard Bernoulli scheme with two possible outcomes.
We can see that solving this problem using the standard Bernoulli scheme will be very problematic since we will have to go through all the possible probabilities in turn, the probability that we hit 50, times hit 51 times, and so on up to 70. That is why we will use the CLT to solve this task.

Task

In the image above, we showed that the value of interest to us can be approximated using a Gaussian distribution with a mean equal to 40 and a variance equal to 24.

Your task is to calculate the required probability: in the first section, we considered that you can use CDF for this. Your task is:

Import norm class from scipy.stats module.
Use .cdf() method of norm class to calculate probability.

Everything was clear?

Section 2. Chapter 5

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: Application of the CLT to Solving Real Problem

Let's imagine that we need to solve the following problem:

Suppose that we come to the shooting range and start shooting, the probability of hitting the target is 0.4, respectively the probability of missing is 0.6;
We shot 100 times and needed to calculate the probability that the hits would be between 50 and 70.

Task

In the image above, we showed that the value of interest to us can be approximated using a Gaussian distribution with a mean equal to 40 and a variance equal to 24.

Your task is to calculate the required probability: in the first section, we considered that you can use CDF for this. Your task is:

Import norm class from scipy.stats module.
Use .cdf() method of norm class to calculate probability.

Everything was clear?

Section 2. Chapter 5