Conditional probability

Conditional expected value

Given an event

Let be a real random variable and be an event with nonzero probability. Then the conditional expected value of given is defined to be prob

where is the conditional distribution of given .

Properties

• If partition then

Given a random variable

Let be a real random variables and . Then the conditional expected value of given is the random function

Properties

Let be real random variables.

1. If and are independent, then
2. For any function we have
3. Linearity:
4. Adam’s law:
5. Adam’s law with extra conditioning:
6. Projection interpretation of conditional expected value

See also

• Conditional variance

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