## 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 and (they sum to one). The division by denomination ($1 or $5) divides the same sample space differently, see the lower right picture, with the probabilities to pull out $1 and $5 equal to and , resp. (they also sum to one). This is summarized in the tables

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 it can be a and $1 and the corresponding probability is The two divisions of the sample space generate a new division into four parts. Then geometrically it is obvious that we have four identities:

*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 taking values and resp., and their own probabilities are Denote the joint probabilities Then we have the identities

(1) ( equations).

**In words**: to obtain the marginal probability of one variable (say, ) sum over the values of the other variable (in this case, ).

The name marginal probabilities is used for because in the two-dimensional table they arise as a result of summing table entries along columns or rows and are displayed in the margins.

### Analogs for continuous variables with densities

Suppose we have two continuous random variables and their own densities are and Denote the joint density . Then replacing in (1) sums by integrals and probabilities by densities we get

(2)

**In words**: to obtain one marginal density (say, ) integrate out the other variable (in this case, ).

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