Definition of Conditional Probability
For any event, define the indicator function:
which is a random variable with respect to the Borel σ-algebra on (0,1). Note that the expectation of this random variable is equal to the probability of A itself:
Then the conditional probability given is a function such that is the conditional expectation of the indicator function for A:
In other words, is a -measurable function satisfying
A conditional probability is regular if is also a probability measure for all ω ∈ Ω. An expectation of a random variable with respect to a regular conditional probability is equal to its conditional expectation.
- For the trivial sigma algebra the conditional probability is a constant function,
- For, as outlined above, .
Read more about this topic: Conditional Expectation
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