On the Relationship between the Conditional and Unconditional Distribution of a Random Variable

This paper studies the relationship between the unconditional and conditional distribution of the same random variable, say Y, when the distribution of the conditioning random variable X is of a known form. The problem is tackled in the general case where the distribution of Y and Y given X are mixed. Attention is focused to two particular cases. In the first X is assumed to follow a Poisson distribution; in the second X is allowed to have a mixed Poisson form. Potential applications are also discussed