One may add that the marginal distributions of  an  relative to their joint distributions are identical with the distributions of  and  taken individually one at a time, i.e.,

and  and  have the properties of the univariate distributions of X and Y respectively.

In a similar way we can define the marginal distributions in the case when we have  number of r.v., , such that each has the required probability space is . Let us also define
,…,  as the probability measure induced by  respectively on the space, defined as , where . If  are r.v., such that they are defined on , where  are arbitrary Borel sets identifined for  then  be the joint distribution function defined for the r.v., , where. Then  represents the probability that the variable  will take a value belonging to the area marked by , i.e., the probability , irrespective of the value of  (remember  can take any value in the entire domain of , i.e., , where ). In a similar way we can define  and one can write , , for some r.v., XI.

Thus the probabilities  for varying values of XI defines the m arginal distribution of X₁ relative to the joint distribution of . In a very simple sense it means we project the mass of the joint distribution on the sub-space of the variable X₁. In a similar sense we can also say that the marginal distributions of XI on  relative to their joint distribution is identical with the distribution of X₁ taken individually one at a time, i.e.,

For a stochastic process we will consider the joint distribution, Thus given we are interested to find the joint distribution of .

Note
We say a stochastic process is a stationary joint distribution if it is invariant to the shift of time, i.e., if the joint distribution of  and  are the same . This is usually called stationary of order n as we have n ordered time points. If a stochastic process is said is said to be of order n for every value of , then the stochastic process is called strictly stationary.