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: Resampling Approach to Compare Mean Values of the Datasets

We can also use the resampling approach to test the hypothesis with non-Gaussian datasets. Resampling is a technique for sampling from an available data set to generate additional samples, each of which is considered representative of the underlying population.

Approach description

Let's describe the most simple resampling method to check main hypothesis that two datasets X and Y have equal mean values:

Concatenate both arrays (X and Y) into one big array;
Shuffle that entire array so observations from each group are spread randomly throughout that array instead of being separated at the breaking point;
Arbitrarily split the array in the breaking point (X_length), assign observations below index len(X_length) to Group A and the rest to Group B;
Subtract the mean of this new Group A from the mean of the new Group B. This would give us one permutation test statistic;
Repeat those steps N times to simulate the main hypothesis distribution;
Calculate test statistics on initial sets X and Y;
Determine critical values of the main hypothesis distribution;
Check if the test statistic calculated on initial sets falls into a critical area of the main hypothesis distribution. If it falls then reject the main hypothesis.

Let's apply this approach in code:

Task

Your task is to implement described above resampling algorithm and to check the corresponding hypothesis on two datasets:

Use the np.concatenate() method to merge X and Y arrays.
Use the .shuffle() method of the np.random module to shuffle data in the merged array.
Use np.quantile() method to calculate left critical value.
Use the created resampling_test() function to check the hypothesis on generated data.

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

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Section 4. Chapter 5

Challenge: Resampling Approach to Compare Mean Values of the Datasets

Approach description

Let's describe the most simple resampling method to check main hypothesis that two datasets X and Y have equal mean values:

Concatenate both arrays (X and Y) into one big array;
Shuffle that entire array so observations from each group are spread randomly throughout that array instead of being separated at the breaking point;
Arbitrarily split the array in the breaking point (X_length), assign observations below index len(X_length) to Group A and the rest to Group B;
Subtract the mean of this new Group A from the mean of the new Group B. This would give us one permutation test statistic;
Repeat those steps N times to simulate the main hypothesis distribution;
Calculate test statistics on initial sets X and Y;
Determine critical values of the main hypothesis distribution;
Check if the test statistic calculated on initial sets falls into a critical area of the main hypothesis distribution. If it falls then reject the main hypothesis.

Let's apply this approach in code:

Task

Your task is to implement described above resampling algorithm and to check the corresponding hypothesis on two datasets:

Use the np.concatenate() method to merge X and Y arrays.
Use the .shuffle() method of the np.random module to shuffle data in the merged array.
Use np.quantile() method to calculate left critical value.
Use the created resampling_test() function to check the hypothesis on generated data.

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 4. Chapter 5

Challenge: Resampling Approach to Compare Mean Values of the Datasets

Approach description

Let's describe the most simple resampling method to check main hypothesis that two datasets X and Y have equal mean values:

Concatenate both arrays (X and Y) into one big array;
Shuffle that entire array so observations from each group are spread randomly throughout that array instead of being separated at the breaking point;
Arbitrarily split the array in the breaking point (X_length), assign observations below index len(X_length) to Group A and the rest to Group B;
Subtract the mean of this new Group A from the mean of the new Group B. This would give us one permutation test statistic;
Repeat those steps N times to simulate the main hypothesis distribution;
Calculate test statistics on initial sets X and Y;
Determine critical values of the main hypothesis distribution;
Check if the test statistic calculated on initial sets falls into a critical area of the main hypothesis distribution. If it falls then reject the main hypothesis.

Let's apply this approach in code:

Task

Your task is to implement described above resampling algorithm and to check the corresponding hypothesis on two datasets:

Use the np.concatenate() method to merge X and Y arrays.
Use the .shuffle() method of the np.random module to shuffle data in the merged array.
Use np.quantile() method to calculate left critical value.
Use the created resampling_test() function to check the hypothesis on generated data.

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

Everything was clear?

Thanks for your feedback!

Approach description

Let's describe the most simple resampling method to check main hypothesis that two datasets X and Y have equal mean values:

Concatenate both arrays (X and Y) into one big array;
Shuffle that entire array so observations from each group are spread randomly throughout that array instead of being separated at the breaking point;
Arbitrarily split the array in the breaking point (X_length), assign observations below index len(X_length) to Group A and the rest to Group B;
Subtract the mean of this new Group A from the mean of the new Group B. This would give us one permutation test statistic;
Repeat those steps N times to simulate the main hypothesis distribution;
Calculate test statistics on initial sets X and Y;
Determine critical values of the main hypothesis distribution;
Check if the test statistic calculated on initial sets falls into a critical area of the main hypothesis distribution. If it falls then reject the main hypothesis.

Let's apply this approach in code:

Task

Your task is to implement described above resampling algorithm and to check the corresponding hypothesis on two datasets:

Use the np.concatenate() method to merge X and Y arrays.
Use the .shuffle() method of the np.random module to shuffle data in the merged array.
Use np.quantile() method to calculate left critical value.
Use the created resampling_test() function to check the hypothesis on generated data.

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

Section 4. Chapter 5

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