14.2 Central limit theorem and its variants

The Central Limit Theorem is a key concept in statistics, showing how sample means tend to follow a normal distribution as sample size grows. It's crucial for making inferences about populations from sample data.

This section explores variations of the theorem, including the law of large numbers and multivariate cases. It also dives into the properties of normal distributions, their applications, and measures of variability.

Central Limit Theorem and its Variants

Fundamental Concepts of Central Limit Theorem

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• Central limit theorem states that the distribution of sample means approximates a normal distribution as the sample size becomes larger
• Applies to independent and identically distributed random variables with finite mean and variance
• Sample size typically considered "large enough" when n ≥ 30
• Enables statistical inference and hypothesis testing for population parameters
• Formula for the distribution of sample means: $\bar{X} \sim N(\mu, \frac{\sigma^2}{n})$
• Allows for the calculation of probabilities and confidence intervals for sample means

Extensions and Variations of the Central Limit Theorem

• Law of large numbers demonstrates that the sample mean converges to the population mean as sample size increases
• Weak law of large numbers states convergence in probability
• Strong law of large numbers states convergence with probability 1
• Berry-Esseen theorem quantifies the rate of convergence to the normal distribution
• Provides an upper bound on the maximum difference between the cumulative distribution function of the sample mean and that of the normal distribution
• Local limit theorem focuses on the convergence of probability mass functions or probability density functions
• Applies to discrete random variables and provides a more precise approximation for specific values

Multivariate Central Limit Theorem

• Extends the central limit theorem to vector-valued random variables
• Considers the joint distribution of multiple random variables
• Assumes independent and identically distributed random vectors with finite mean vector and covariance matrix
• Resulting distribution is a multivariate normal distribution
• Covariance matrix of the sample mean is scaled by 1/n, where n is the sample size
• Enables analysis of relationships between multiple variables in large samples

Normal and Gaussian Distributions

Properties and Characteristics of Normal Distribution

• Normal distribution characterized by a symmetric, bell-shaped curve
• Also known as Gaussian distribution, named after Carl Friedrich Gauss
• Probability density function given by: $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$
• Defined by two parameters: mean (μ) and standard deviation (σ)
• Mean determines the center of the distribution
• Standard deviation determines the spread or width of the distribution
• Approximately 68% of data falls within one standard deviation of the mean
• Approximately 95% of data falls within two standard deviations of the mean
• Approximately 99.7% of data falls within three standard deviations of the mean (empirical rule)

Applications and Transformations of Normal Distribution

• Standard normal distribution has a mean of 0 and standard deviation of 1
• Z-score represents the number of standard deviations a data point is from the mean
• Formula for Z-score: $Z = \frac{X - \mu}{\sigma}$
• Z-scores allow for comparison of values from different normal distributions
• Many natural phenomena follow a normal distribution (height, weight, IQ scores)
• Central limit theorem explains why many real-world distributions approximate normal distribution
• Logarithmic transformations can sometimes convert skewed distributions to approximately normal distributions

Measures of Variability in Normal Distribution

• Variance measures the average squared deviation from the mean
• Calculated as the average of squared differences between each data point and the mean
• Formula for variance: $\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n}$
• Standard deviation is the square root of variance
• Provides a measure of spread in the same units as the original data
• Formula for standard deviation: $\sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n}}$
• Coefficient of variation (CV) expresses standard deviation as a percentage of the mean
• Useful for comparing variability between datasets with different units or scales