Зміст курсу

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: Checking Bias of An Estimation Using Simulation

In the last chapter, we covered the concepts of sample variance and adjusted sample variance. Now let's see how with the help of simulation, we can determine that the first estimation is biased and the second is unbiased.

We will use the Gaussian population: we will build an estimate of the sample variance and the adjusted sample variance on different subsets of the population. Next, using the law of large numbers, we will estimate the mean of the sample variance and the adjusted sample variance and compare it with the real variance of the population.

Завдання

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Your task is to perform simulations to obtain the value of the sample variance, and the adjusted sample variance for 2000 different subsets of the population and compare the mean of the sample variance and the adjusted sample variance with the real value of the population mean:

Use ddof=0 as an argument of np.var() method to calculate sample variance.
Use ddof=1 as an argument of np.var() method to calculate the adjusted sample variance.
Use .mean() method to estimate the expectation of sample variance.

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