Marginal probabilities and densities
This is to help everybody, from those who study Basic Statistics up to Advanced Statistics ST2133.
Discrete case
Suppose in a box we have coins and banknotes of only two denominations: $1 and $5 (see Figure 1).

Figure 1. Illustration of two variables
We pull one out randomly. The division of cash by type (coin or banknote) divides the sample space (shown as a square, lower left picture) with probabilities
Variable 1: Cash type | Prob |
coin | |
banknote |
Variable 2: Denomination | Prob |
$1 | |
$5 |
Now we can consider joint events and probabilities (see Figure 2, where the two divisions are combined).


For example, if we pull out a random
Adding over denominations:
Adding over cash types:
Formally, here we use additivity of probability for disjoint events
In words: we can recover own probabilities of variables 1,2 from joint probabilities.
Generalization
Suppose we have two discrete random variables
(1)
In words: to obtain the marginal probability of one variable (say,
The name marginal probabilities is used for
Analogs for continuous variables with densities
Suppose we have two continuous random variables
(2)
In words: to obtain one marginal density (say,
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