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: Confidence Interval for Exponential Distribution Parameter

In the previous chapter, we considered how to build confidence intervals for the parameters of a Gaussian distribution. Now let's understand how we can use this to build confidence intervals for the parameters of other distributions.
For example, let's use exponential distribution. Earlier in this course, the exponential distribution has already been used in examples. Now, let's look at this distribution in more detail and consider its PDF and characteristics:

We have to build a confidence interval for the lambda parameter to solve this problem. Let's first consider not the lambda parameter itself but the inversed parameter 1/lambda.
According to the method of moments, we can estimate this parameter using the sample mean, as we have shown earlier, such an estimate will be both non-biased and consistent.

Note

This estimate will also be effective You can verify this yourself based on the information in the previous chapter.

Confidence interval for exponential distribution parameter

Firstly, let's find the expected value and variance of our estimation:

Now let's construct the following function and analyze it:

Since we have the sum of i.i.d. random variables with a finite mathematical expectation and variance, we can apply the central limit theorem to our estimate. Then we construct the function so that its distribution does not depend on the estimated parameters. Finally, we only have to specify the analytical confidence interval.

Note

Estimates for which the Central Limit Theorem holds are called asymptotically normal. We can construct confidence intervals using the confidence intervals of the Gaussian distribution for such estimates.

Task

Swipe to show code editor

Thus, we analytically derived a formula by which we can calculate the confidence interval for the exponential distribution parameter, it remains to implement this in the code. You have to:

Import stats module from scipy library to provide necessary calculations.
Use .ppf() method to calculate z_value.
Use .mean() method to get estimation of 1 / lambda parameter.

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

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Thanks for your feedback!

Section 3. Chapter 9

Challenge: Confidence Interval for Exponential Distribution Parameter

Note

This estimate will also be effective You can verify this yourself based on the information in the previous chapter.

Confidence interval for exponential distribution parameter

Firstly, let's find the expected value and variance of our estimation:

Now let's construct the following function and analyze it:

Note

Estimates for which the Central Limit Theorem holds are called asymptotically normal. We can construct confidence intervals using the confidence intervals of the Gaussian distribution for such estimates.

Task

Swipe to show code editor

Thus, we analytically derived a formula by which we can calculate the confidence interval for the exponential distribution parameter, it remains to implement this in the code. You have to:

Import stats module from scipy library to provide necessary calculations.
Use .ppf() method to calculate z_value.
Use .mean() method to get estimation of 1 / lambda parameter.

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 3. Chapter 9

Challenge: Confidence Interval for Exponential Distribution Parameter

Note

This estimate will also be effective You can verify this yourself based on the information in the previous chapter.

Confidence interval for exponential distribution parameter

Firstly, let's find the expected value and variance of our estimation:

Now let's construct the following function and analyze it:

Note

Estimates for which the Central Limit Theorem holds are called asymptotically normal. We can construct confidence intervals using the confidence intervals of the Gaussian distribution for such estimates.

Task

Swipe to show code editor

Thus, we analytically derived a formula by which we can calculate the confidence interval for the exponential distribution parameter, it remains to implement this in the code. You have to:

Import stats module from scipy library to provide necessary calculations.
Use .ppf() method to calculate z_value.
Use .mean() method to get estimation of 1 / lambda parameter.

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

Everything was clear?

Thanks for your feedback!

Note

This estimate will also be effective You can verify this yourself based on the information in the previous chapter.

Confidence interval for exponential distribution parameter

Firstly, let's find the expected value and variance of our estimation:

Now let's construct the following function and analyze it:

Note

Estimates for which the Central Limit Theorem holds are called asymptotically normal. We can construct confidence intervals using the confidence intervals of the Gaussian distribution for such estimates.

Task

Swipe to show code editor

Thus, we analytically derived a formula by which we can calculate the confidence interval for the exponential distribution parameter, it remains to implement this in the code. You have to:

Import stats module from scipy library to provide necessary calculations.
Use .ppf() method to calculate z_value.
Use .mean() method to get estimation of 1 / lambda parameter.

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

Section 3. Chapter 9

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