1. Conditional Variance

#4-oct

Definition (Conditional Variance).

Theorem.

Proof.

Also as

by adding (6.1) and (6.2) we get

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Important

Refer 7.5.4 Conditional Variance from A first course in Probability by - Sheldon Ross (page number: 328)
Example 5o 5p which may be done in class but not written in this notes. Don't worry they are about the same

#9-oct

Example

  1. independent and identical RV
    is a non-negative integer valued RV which is independent of

Predictor of Y

When we know and what to find what is the we may chose to be a predictor of given . A good way of judging the closeness of is to minimize

Theorem.

is the closest predictor of in other words
for all

Proof.

as is a function of when is given it can be considered as a constant

hence

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