Visualizing a bivariate discrete distribution and other distributions derived from it

A single discrete random variable is depicted by a stick diagram, a 2D picture. Naturally, to visualize a bivariate discrete distribution, one can use a bivariate stick diagram, a 3D picture. Unfortunately, many students have difficulty understanding and processing 3D pictures. Therefore, we construct an alternative 2D disc plot to depict the bivariate distribution of (, ), from which we obtain graphically the conditional distributions of given , and given ; the marginal distributions of , , , . Furthermore, we depict the mean and the standard deviation of each distribution using a single‐headed arrow. We hope these visualizations will help students better comprehend these concepts and avoid some misconceptions.