Conditional Distributions

Conditional Probability Mass Function

Definition

The conditional probability mass function is

if .

Conditional Probability Density Function

For continuous distributions, we use the same definition as the above. But one must be careful. When we compute in the continuous case, we are conditioning on the event , which as probability 0. We avoid this problem by defining things in terms of the PDF. The fact that this leads to a well-defined theory is left for more advanced courses.

Definition

For continuous random variables, the conditional probability density function is

assuming that . Then,

From the definition of the conditional density, we see that .

Sources

• Wasserman, L. (2010). All of Statistics: A concise Course in Statistical Inference. Chapter 2.8.