Probability and Distributions for Environmental and Health Data

This chapter, “Probability and Distributions for Environmental and Health Data,” provides a theoretical foundation for understanding uncertainty and variability in environmental health data. It begins with probability, random variables, sample spaces, conditional probability, independence, and Bayes’ theorem. Students are introduced to probability mass functions (PMF), probability density functions (PDF), and cumulative distribution functions (CDF), with practical examples involving PM2.5 concentrations and daily respiratory disease cases. The chapter then develops expected value and variance as fundamental measures of a random variable's center and variability, emphasizing their importance in environmental exposure assessment, risk analysis, parameter estimation, and statistical modeling.

The chapter subsequently examines several probability distributions that are particularly relevant to environmental health statistics. The normal distribution, along with the standardized Z distribution and t distribution, forms the basis for probability calculations, confidence intervals, and hypothesis testing. Because environmental exposure data are often right-skewed, the chapter also discusses the lognormal and gamma distributions, including logarithmic transformation and their applications to pollutant concentrations and biological biomarkers. For binary outcomes, the chapter introduces the binomial distribution, while it presents the Poisson and negative binomial distributions for count data such as disease cases and vector counts. Particular attention is given to overdispersion and the use of Negative Binomial models when the variance exceeds the mean. Examples in R show how to apply these distributions to real-world environmental health problems.

Finally, the chapter connects probability distributions with statistical inference and dependence structures in environmental health research. The Central Limit Theorem (CLT) explains why sample means can approximately follow a normal distribution even when the underlying population is non-normal, providing the foundation for standard errors, confidence intervals, and hypothesis testing. This is particularly important for environmental measurements that are frequently skewed, such as arsenic, PM2.5, and other pollutant concentrations. The chapter then introduces correlation and dependence, including covariance, Pearson and Spearman correlations, partial correlation, temporal autocorrelation, and spatial autocorrelation using measures such as Moran’s I. These concepts are essential because environmental and health observations may be correlated across time and geographic locations; ignoring such dependence can lead to underestimated standard errors and invalid statistical inference. The course material and homework click the link here (Material, Tugas I, dan Tugas II)