## 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).

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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