The Central Limit Theorem states that, given certain conditions, the distribution of the mean of a large sample of independent and identically distributed (i.i.d.) random variables will approximate a normal distribution, even if the original variables are not normally distributed. In other words, as the sample size (n) increases, the distribution of the sample mean (x̄) converges to a normal distribution with mean μ and variance σ²/n.
Key components:
- Independence: The random variables must be independent of each other.
- Identical distribution: The random variables must have the same distribution.
- Large sample size: The sample size (n) must be sufficiently large.
Properties:
- Asymptotic normality: The distribution of the sample mean converges to a normal distribution as the sample size increases.
- Location parameter: The mean of the sample mean converges to the population mean (μ).
- Scale parameter: The variance of the sample mean converges to σ²/n.
Law of Large Numbers (LLN)
The Law of Large Numbers states that, as the sample size (n) increases, the average of a sequence of independent and identically distributed (i.i.d.) random variables will converge to the population mean (μ) with probability 1.
Key components:
- Independence: The random variables must be independent of each other.
- Identical distribution: The random variables must have the same distribution.
- Large sample size: The sample size (n) must be sufficiently large.
Properties:
- Convergence: The average of the sample converges to the population mean (μ) with probability 1.
- Rate of convergence: The LLN provides no information about the rate at which the sample average converges to the population mean.
Relationship between CLT and LLN
- Refinement: The CLT refines the LLN by providing information about the distribution of the sample mean, whereas the LLN only guarantees convergence to the population mean.
- Assumptions: Both theorems require independence and identical distribution, but the CLT also assumes a large sample size, whereas the LLN does not.
- Applications: The CLT is useful for statistical inference and hypothesis testing, while the LLN is useful for understanding the long-run behavior of averages.
Examples and Applications
- Finance: The CLT is used to model stock prices and portfolio returns, while the LLN is used to understand the long-run behavior of investment returns.
- Quality control: The LLN is used to monitor and control the quality of manufactured products, while the CLT is used to estimate the mean and variance of quality metrics.
- Survey research: The CLT is used to analyze survey data and estimate population means, while the LLN is used to understand the long-run behavior of survey responses.