Appendix C — Statistics

C.1 Marginal distribution

Imagine you have a box of objects of different colors and shapes:

• 3 red balls
• 6 green balls
• 1 red square
• 4 green squares

Let:

• \(X\) is the color of an object (red or green),
• \(Y\) is the shape of an object (ball or square)

If you pick an object at random, the joint distribution \(\mathbb{P}(X, Y)\) describes the probability of every (color, shape) combination:

• 3 red balls
• 6 green balls
• 1 red square
• 4 green squares

• \(\mathbb{P}(X = \text{red}, Y = \text{ball})\)
• \(\mathbb{P}(X = \text{green}, Y = \text{ball})\)
• \(\mathbb{P}(X = \text{red}, Y = \text{square})\)
• \(\mathbb{P}(X = \text{green}, Y = \text{square})\)

If you only care about color, the marginal distribution of color \(\mathbb{P}(X)\), is found by summing the joint probabilities over all shapes:

\[\begin{align} \mathbb{P}(X = \text{red}) &= \mathbb{P}(X = \text{red}, Y = \text{ball}) + \mathbb{P}(X = \text{red}, Y = \text{square}) \\ &= \sum_{y} \mathbb{P}(X = x, Y = y) \end{align}\]