# Difference between revisions of "Marginal distribution"

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For a continuous [[probability density function]] (pdf), an associated marginal pdf can be written as ''m''<sub>''y''</sub>(''x''). Such that | For a continuous [[probability density function]] (pdf), an associated marginal pdf can be written as ''m''<sub>''y''</sub>(''x''). Such that | ||

− | :<math>m_{y}(x) = \int_y p(x,y) \, dy = \int_y c(x|y) \, p(y) \, dy </math> | + | ::<math>m_{y}(x) = \int_y p(x,y) \, dy = \int_y c(x|y) \, p(y) \, dy </math> |

where ''p''(''x'',''y'') gives the [[joint probability density function]] of ''x'' and ''y'', and ''c''(''x''|''y'') gives the [[conditional probability density function]] for ''x'' given ''y''. The second integral was formulated by use of the [[Bayesian product rule]]. Note that the marginal distribution has the form of an [[expectation value]]. | where ''p''(''x'',''y'') gives the [[joint probability density function]] of ''x'' and ''y'', and ''c''(''x''|''y'') gives the [[conditional probability density function]] for ''x'' given ''y''. The second integral was formulated by use of the [[Bayesian product rule]]. Note that the marginal distribution has the form of an [[expectation value]]. | ||

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For a discrete [[probability mass function]] (pmf), the marginal probability for x<sub>k</sub> can be written as ''p''<sub>''k''</sub> Such that | For a discrete [[probability mass function]] (pmf), the marginal probability for x<sub>k</sub> can be written as ''p''<sub>''k''</sub> Such that | ||

− | :<math>p_{k} = \sum_{j} p_{kj} = \sum_{j} p_{j}p_{k|j} </math> | + | ::<math>p_{k} = \sum_{j} p_{kj} = \sum_{j} p_{j}p_{k|j} </math> |

where the ''j'' index spans all indices of the discrete ''y''. The notation ''p''<sub>''kj''</sub> here means the joint probability value when ''x'' has the value ''x''<sub>k</sub> and ''y'' has the value ''y''<sub>j</sub> while ''p''<sub>''k|j''</sub> here references the conditional probability value for ''x''<sub>k</sub> for y fixed at the value ''y''<sub>j</sub>. With ''k'' fixed in the above summation and ''p''<sub>''k'',''j''</sub> considered as a matrix, this can be thought of as summing over all columns in the k<sup>th</sup> row. Similarly, the marginal mass function for ''y''<sub>j</sub> (say ''q''<sub>''j''</sub>) can be computed by summing over all rows in column ''j''. When all of the ''p''<sub>''k''</sub> are determined this way for all k, this set of ''p''<sub>''k''</sub> constitute the pmf for the all relevant discrete values of ''x'', in this particular case calculated as a marginal mass function from an original joint probability mass function. | where the ''j'' index spans all indices of the discrete ''y''. The notation ''p''<sub>''kj''</sub> here means the joint probability value when ''x'' has the value ''x''<sub>k</sub> and ''y'' has the value ''y''<sub>j</sub> while ''p''<sub>''k|j''</sub> here references the conditional probability value for ''x''<sub>k</sub> for y fixed at the value ''y''<sub>j</sub>. With ''k'' fixed in the above summation and ''p''<sub>''k'',''j''</sub> considered as a matrix, this can be thought of as summing over all columns in the k<sup>th</sup> row. Similarly, the marginal mass function for ''y''<sub>j</sub> (say ''q''<sub>''j''</sub>) can be computed by summing over all rows in column ''j''. When all of the ''p''<sub>''k''</sub> are determined this way for all k, this set of ''p''<sub>''k''</sub> constitute the pmf for the all relevant discrete values of ''x'', in this particular case calculated as a marginal mass function from an original joint probability mass function. | ||

[[Category:mathematics]] | [[Category:mathematics]] |

## Revision as of 10:41, 13 December 2007

In probability theory, given a joint probability density function of two parameters or variables *x* and *y*, the **marginal distribution** of *x* is the probability density function of *x* after information about *y* has been averaged out. For example from a Bayesian probability perspective, when doing parameter estimation we can consider the joint probability density function as a joint inference which characterizes our uncertainty about the true values of the two parameters, and the marginal distribution of (say) *x* as our inference about *x* after the uncertainty about *y* had been averaged out. We can say that, in this case, we are considering *y* as a nuisance parameter.

For a continuous probability density function (pdf), an associated marginal pdf can be written as *m*_{y}(*x*). Such that

where *p*(*x*,*y*) gives the joint probability density function of *x* and *y*, and *c*(*x*|*y*) gives the conditional probability density function for *x* given *y*. The second integral was formulated by use of the Bayesian product rule. Note that the marginal distribution has the form of an expectation value.

For a discrete probability mass function (pmf), the marginal probability for x_{k} can be written as *p*_{k} Such that

where the *j* index spans all indices of the discrete *y*. The notation *p*_{kj} here means the joint probability value when *x* has the value *x*_{k} and *y* has the value *y*_{j} while *p*_{k|j} here references the conditional probability value for *x*_{k} for y fixed at the value *y*_{j}. With *k* fixed in the above summation and *p*_{k,j} considered as a matrix, this can be thought of as summing over all columns in the k^{th} row. Similarly, the marginal mass function for *y*_{j} (say *q*_{j}) can be computed by summing over all rows in column *j*. When all of the *p*_{k} are determined this way for all k, this set of *p*_{k} constitute the pmf for the all relevant discrete values of *x*, in this particular case calculated as a marginal mass function from an original joint probability mass function.