Overview — Sampling Distributions and the Normal Family

This topic builds the toolkit. It derives the three workhorses of normal-theory inference — the chi-squared ($\chi^2$), Student's $t$, and Snedecor's $F$ distributions — from the normal distribution, and shows how the sample mean and sample variance land exactly in their respective families. Along the way it extends the univariate sample-statistic machinery to the bivariate case (sample covariance and correlation), introduces order statistics, which generalise the median and quantiles to a fully probabilistic treatment, and covers the empirical CDF together with QQ-plots, the practitioner's first line of defence for checking whether a distributional assumption is plausible.

By the end of this topic you should be able to:

• state and apply the sampling distributions of $\bar{X}$ and $S^2$, and the standard error, for a sample from any population;
• compute sample covariance and sample correlation, and distinguish cross-sectional dispersion from time-series volatility;
• state the definition of the $\chi^2$, $t$, and $F$ distributions and write down their construction from normals;
• derive that $(n-1)S^2/\sigma^2 \sim \chi^2(n-1)$ and that $\bar{X}$ and $S^2$ are independent under normality;
• compute tail probabilities for $t$ and $F$ random variables and identify which test statistic follows which distribution;
• find the density of the $k$-th order statistic and use it to characterise sample quantiles;
• construct an ECDF and read a QQ-plot to diagnose departures from a distributional assumption.

1. Contents
2. The big picture