45 Marginalisation

Marginal distribution: A distribution formed by calculating the subset of a larger probability distribution of a collection of random variables.

Example

Two random variables X and Y with joint PMF and marginal

PMFs:

$$\begin{array}{ccccc}& Y=0& Y=1& Y=2& P_X(x)\\ X=0& \frac{4}{24}& \frac{6}{24}& \frac{3}{24}& \frac{13}{24}\\ X=1& \frac{3}{24}& \frac{4}{24}& \frac{4}{24}& \frac{11}{24}\\ P_Y(y)& \frac{7}{24}& \frac{10}{24}& \frac{7}{24}& 1\end{array}$$

Calculation :

• For discrete random variables, we sum over the unwanted variable.
• For continuous random variables, we would integrate over the unwanted variable.