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

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.

Switch to desktop for real-world practiceContinue from where you are using one of the options below

Everything was clear?

Thanks for your feedback!

Section 2. Chapter 5

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.

Switch to desktop for real-world practiceContinue from where you are using one of the options below

Everything was clear?

Thanks for your feedback!

Section 2. Chapter 5

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.

Switch to desktop for real-world practiceContinue from where you are using one of the options below

Everything was clear?

Thanks for your feedback!

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.

Switch to desktop for real-world practiceContinue from where you are using one of the options below

Section 2. Chapter 5

Switch to desktop for real-world practiceContinue from where you are using one of the options below