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: Using CLT to Compare Mean Values of Non-Gaussian Datasets

In the last chapter, we considered how to compare the mathematical expectations of two Gaussian datasets. But what if the datasets are not Gaussian, and is it possible to somehow compare them in this case?

Using Central Limit Theorem to compare mean values

We can use the CLT to compare mean values of non-Gaussian datasets:

If we have many samples, we can use the CLT to construct new features: instead of analyzing samples, we can analyze the mean values of the samples. Due to CLT, if we calculate the mean with many samples, this mean value will be normally distributed;
Use the Student criterion described in the previous chapter to test the hypothesis.

Note

For different distributions, you need to select a different number of samples for which the average is calculated to achieve normality. This is usually done experimentally using various tests for normality, for example, shapiro normality test.

Task

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Now we will check the hypothesis that two exponential datasets have equal mean values using the Central Limit Theorem. Your task is:

Import ttest_ind function from scipy.stats module to provide t-test.
Use .mean() method to calculate the mean over the sliding window in sliding_mean function.
Use shapiro() function to check normality of X_mean array.
Specify condition in if statement to check hypothesis.

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Section 4. Chapter 4

Challenge: Using CLT to Compare Mean Values of Non-Gaussian Datasets

Using Central Limit Theorem to compare mean values

We can use the CLT to compare mean values of non-Gaussian datasets:

If we have many samples, we can use the CLT to construct new features: instead of analyzing samples, we can analyze the mean values of the samples. Due to CLT, if we calculate the mean with many samples, this mean value will be normally distributed;
Use the Student criterion described in the previous chapter to test the hypothesis.

Note

For different distributions, you need to select a different number of samples for which the average is calculated to achieve normality. This is usually done experimentally using various tests for normality, for example, shapiro normality test.

Task

Swipe to show code editor

Now we will check the hypothesis that two exponential datasets have equal mean values using the Central Limit Theorem. Your task is:

Import ttest_ind function from scipy.stats module to provide t-test.
Use .mean() method to calculate the mean over the sliding window in sliding_mean function.
Use shapiro() function to check normality of X_mean array.
Specify condition in if statement to check hypothesis.

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 4

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