The following proposition provides an answer to this question.
Discrete Distributions Section 3: Definition Let be a discrete random vector. The impossibility of deriving the conditional probability mass function unambiguously in this case called by some authors the Borel-Kolmogorov paradox is not particularly worrying, as this case is seldom relevant in applications.
Suppose X and Y are continuous random variables with joint probability density function f x , y and marginal probability density functions f X x and f Y y , respectively. Unable to display preview.
The following is an example of a case in which the conditional probability mass function cannot be derived unambiguously the example is a bit involved; the reader might safely skip it on a first reading. Printer-friendly version Thus far, all of our definitions and examples concerned discrete random variables, but the definitions and examples can be easily modified for continuous random variables.
It means that any choice of is legitimate, provided the requirement is satisfied. One can show that also the requirement that be a regular conditional probability does not help to pin down.Marginal distribution and conditional distribution - AP Statistics - Khan Academy
Thus, for we trivially have because , while for we have Thus, the marginal probability density function of is. Cite chapter How to cite?
The fundamental property of conditional probability is satisfied in this case if and only if, for a given , the following system of equations is satisfied: Eberly College of Science. In this case, the partition of interest is , where and can be viewed as the realization of the conditional probability when. The support of the vector is and the joint probability function of and is The marginal probability density function of is obtained by marginalization, integrating out of the joint probability density function Thus, for we trivially have because , while for we have Thus, the marginal probability density function of is.
We can use the formula: Here's what the joint support S looks like: Bayesian Methods Section 10: In general, when is neither discrete nor absolutely continuous, we can characterize the distribution function of conditional on the information that. This is just the usual formula for computing conditional probabilities conditional probability equals joint probability divided by marginal probability: Example Suppose we are given the following sample space: The Correlation Coefficient Lesson 19: