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: 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.

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Section 2. Chapter 3

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.

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 3

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.

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

Everything was clear?

Thanks for your feedback!

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.

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

Section 2. Chapter 3

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