7.4 Applications of limit theorems

Limit theorems are the backbone of probability theory, shaping how we understand large-scale events. They help us make sense of complex systems by revealing patterns that emerge as sample sizes grow, bridging the gap between theory and real-world applications.

In this section, we'll see how these powerful tools are used in fields like insurance, finance, and quality control. We'll explore how they enable statistical inference, support hypothesis testing, and provide a foundation for many common statistical techniques.

Applying Limit Theorems

Law of Large Numbers Applications

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• Law of large numbers states sample mean converges to population mean as sample size increases
• Applied in insurance for estimating expected claim amounts (informs premium calculations)
• Used in gambling to predict long-term outcomes of games (roulette, slot machines)
• Employed in quality control to estimate defect rates in large production runs
• Supports portfolio diversification in finance by reducing unsystematic risk
  • Adding uncorrelated assets decreases overall portfolio volatility
  • Example: Combining stocks from different sectors (technology, healthcare, energy)

Central Limit Theorem Applications

• Central limit theorem asserts sample means distribution approaches normal as sample size increases
• Crucial in economics for analyzing aggregate economic indicators (GDP, inflation rates)
• Applied in psychology to study population-level behavioral trends
• Used in epidemiology to estimate disease prevalence and incidence rates
• Enables manufacturing quality control by estimating probabilities of defects
  • Example: Predicting likelihood of faulty components in large batches of electronics
• Utilized in political polling to determine sample sizes and calculate margins of error
  • Example: Estimating voter preferences with specified confidence levels

Statistical Inference and Hypothesis Testing

• Both theorems enable population inferences based on sample data
• Form foundation for statistical hypothesis tests (t-tests, ANOVA)
• Support construction of confidence intervals for parameter estimation
• Allow researchers to quantify uncertainty in statistical estimates
• Enable calculation of p-values in hypothesis testing procedures
• Justify use of z-scores and standard normal tables in statistical analysis

Normal Approximations in Inference

Sampling Distributions

• Central limit theorem justifies normal approximations for sampling distributions of means
• Valid even when underlying population is not normally distributed
• Sampling distribution of proportions approximated by normal distribution for large samples
• Enables use of z-tests for hypothesis testing with large sample sizes
• Supports construction of confidence intervals for population parameters
  • Example: Estimating average height of a population based on a sample

Approximating Discrete Distributions

• Normal approximation to binomial distribution valid for large samples and non-extreme probabilities
• Typically applied when np > 5 and n(1-p) > 5 (n = sample size, p = probability of success)
• Continuity correction improves accuracy when approximating discrete distributions
  • Adjusts for discrepancy between continuous normal and discrete binomial distributions
• Enables simplified calculations for binomial probabilities with large n
  • Example: Estimating probability of observing a certain number of defective items in a large batch

Robustness of Parametric Tests

• Many parametric tests robust to normality assumption violations due to central limit theorem
• T-tests remain valid for moderately non-normal distributions with larger sample sizes
• ANOVA (Analysis of Variance) maintains reliability for slight departures from normality
• Enables broader application of these tests in real-world scenarios with imperfect data
• Justifies use of parametric methods even when population distribution is unknown
  • Example: Comparing mean test scores across multiple student groups

Foundation of Statistical Techniques

Confidence Intervals and Hypothesis Testing

• Central limit theorem underpins construction of confidence intervals
• Allows quantification of uncertainty in parameter estimates
• Supports z-tests and t-tests for hypothesis testing
• Enables calculation of p-values for statistical inference
• Justifies use of normal distribution in many statistical procedures
  • Example: Constructing 95% confidence interval for population mean

Regression and Maximum Likelihood Estimation

• Ordinary least squares regression assumes normality of residuals based on central limit theorem
• Asymptotic properties of maximum likelihood estimators derived from limit theorems
• Supports validity of regression coefficients and their standard errors
• Enables inference in generalized linear models and other advanced statistical techniques
• Justifies use of normal approximations in logistic regression for large samples
  • Example: Estimating effect of advertising spend on sales using linear regression

Resampling and Bayesian Methods

• Bootstrap resampling methods justified by law of large numbers and central limit theorem
• Allows estimation of sampling distributions for complex statistics
• Bayesian inference often relies on asymptotic normality results from limit theorems
• Supports derivation of credible intervals in Bayesian analysis
• Enables approximation of posterior distributions in large sample scenarios
  • Example: Using bootstrap to estimate confidence intervals for median income

Interpreting Limit Theorem Results

Insurance and Risk Assessment

• Interpret law of large numbers results as convergence of average claim amounts to expected values
• Use to inform premium calculations and risk assessment strategies
• Explain how increasing policyholder base improves prediction accuracy
• Quantify expected long-term profitability of insurance products
• Assess impact of catastrophic events on overall risk portfolio
  • Example: Estimating average annual flood damage claims for a region

Manufacturing and Quality Control

• Interpret central limit theorem applications as probabilities of specific defect rates
• Guide decision-making on process adjustments and quality improvements
• Explain reliability of sampling inspections for large production runs
• Quantify expected variation in product characteristics
• Assess impact of changes in manufacturing processes on product quality
  • Example: Predicting likelihood of meeting quality standards for semiconductor chip production

Financial Markets and Portfolio Management

• Explain how law of large numbers supports reduction of unsystematic risk through diversification
• Quantify benefits of holding varied portfolio in terms of risk reduction
• Interpret results in context of expected returns and volatility
• Assess impact of adding new assets to existing portfolios
• Explain limitations of diversification for systematic market risk
  • Example: Calculating expected risk reduction from increasing number of stocks in a portfolio from 10 to 50

Medical Research and Clinical Trials

• Interpret limit theorem applications in terms of reliability and generalizability of trial outcomes
• Explain how sample size affects confidence in results and statistical power
• Assess probability of observing treatment effects in larger populations
• Quantify uncertainty in estimated treatment efficacy
• Explain importance of replication and meta-analysis in context of limit theorems
  • Example: Interpreting results of a drug efficacy study with 1000 participants