# 7

# Conditional Probability Models

In many applications of probability, we have a probability model of an experiment but it is impossible to observe the outcome of the experiment. Instead we observe an event that is related to the outcome. In some applications, the outcome of interest, for example a sample value of random voltage *X*, can be obscured by random noise *N*, and we observe only a sample value of *X* + *N*. In other examples, we obtain information about a random variable before it is possible to observe the random variable. For example, we might learn the nature of an email (whether it contains images or only text) before we observe the number of bytes that need to be transmitted. In another example, we observe that the beginning of a lecture is delayed by two minutes and we want to predict the actual starting time. In these situations, we obtain a conditional probability model by modifying the original probability model (for the voltage, or the email size, or the starting time) to take into account the information gained from the event we have observed.

## 7.1 Conditioning a Random Variable by an Event

The conditional PMF *P*_{X|B}(*x*) and conditional PDF *P*_{X|B}(*x*) are probability models that use the definition of conditional probability, Definition 1.5, to incorporate partial knowledge of the outcome of an experiment. The partial knowledge is that the outcome is *X* *B* ⊂ *S*_{X}.

Recall from ...