Marginal Distributions

Let us again define two r.v., ,  such that the required probability space is . Let us also define  and  as the probability measure induced by  and  respectively on the space, defined by..Now if  are r.vs, such that they are defined on , where  and  are arbitrary Borel sets defined for  and  and  be the joint distribution function defined for the r.v., , where  is true,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 Y, i.e., ). In a similar way we can define . We can write  and .

Thus the probabilities  (Figure 1.9) for varying values of  defines the marginal distribution of  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 . Similarly we can define the probabilities  (Figure 1.10) for varying values of such that it defines the marginal distribution of  relative to the joint distribution of  and it implies that we project the mass of the joint distribution on the sub-space of the variable .

Figure 1.9: Illustration of

Figure 1.10: Illustration of