6.10 Sequential Random Realization

Let us write here the details of the decomposition of a joint probability density as a product of one-dimensional marginals and conditionals.

Let us apply a first time the partition of a joint probability density as the product of a marginal and a conditional. Defining

and

gives

Let us apply the partition again. Defining

and

gives

and, with this, equation (6.103) can be written

Continuing this procedure, one arrives at

expressing the joint probability density as a product of different conditional probability densities.

This immediately suggests a method for generating a random point that only uses one-dimensional random generations. One first generates a random value for x ¹ using the unconditional (marginal) density f ₁ (x ¹ ). This gives some value, say . Given this value, one then generates a random value for x ² using the conditional probability density . This gives some value, say . Then, one generates a random value for x ³ using , and so on until one generates a random value for x ⁿ using .