Plotting and Analysis of Three Major Probability Distributions: t, F, and Chi-Square

In statistics, the t-distribution, F-distribution, and chi-square distribution are three crucial probability distributions widely applied in hypothesis testing, analysis of variance, and other statistical domains. By visualizing the shape variations of these distributions under different parameters, we can gain more intuitive understanding of their statistical characteristics. Implementing these visualizations through programming languages like MATLAB or Python allows researchers to dynamically explore parameter effects using distribution functions such as tinv, finv, and chi2inv.

The t-distribution is commonly used for mean inference with small samples, where its shape is significantly influenced by degrees of freedom. With lower degrees of freedom, the curve exhibits thicker tail characteristics; as degrees of freedom increase, the t-distribution gradually approximates the standard normal distribution. Code implementation typically involves generating t-distribution curves using functions like tpdf() while varying the df parameter to observe morphological changes.

The F-distribution is primarily employed in analysis of variance, with its shape determined by two degrees of freedom parameters. Different combinations of numerator and denominator degrees of freedom significantly affect the curve's skewness and peak position. Programming implementation requires specifying both df1 and df2 parameters in F-distribution functions (e.g., fpdf()) to generate comparative plots showing how parameter interactions influence distribution shapes.

The chi-square distribution is frequently used in independence tests and goodness-of-fit tests, with its morphology changing with degrees of freedom. At low degrees of freedom, it displays right-skewed characteristics, gradually becoming more symmetric as degrees of freedom increase. Implementation involves using chi-square distribution functions (chi2pdf()) with progressive df values to demonstrate the transition from skewed to symmetric distributions.

Comparative curves plotted through MATLAB/Python clearly demonstrate: 1) How parameter variations affect distribution curve kurtosis and skewness 2) Morphological differences among distributions 3) Convergence characteristics under limiting conditions Key visualization techniques include overlaying multiple distribution plots with different parameters using hold on/off commands and customizing line styles for clear differentiation.

This visual analysis method provides intuitive tools for understanding statistical distributions, particularly helping students and researchers grasp how parameter changes affect distribution properties. The code-based approach enables interactive exploration through slider controls or parameter arrays, facilitating deeper insights into distribution behavior under varying statistical conditions.