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What is left after conditioning

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$X$ and $Y$ are jointly normal with correlation $0.8$ , and $Y$ has standard deviation $5$ . You learn the value of $X$ . What is the standard deviation of $Y$ given $X$ ? Give a number.

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Hint

The conditional variance is $Var (Y) \times (1 - ρ^{2})$ , and it does not depend on the value you saw.

Solution

The conditional variance is $25 \times (1 - 0. 8^{2}) = 9$ .
Its square root is the standard deviation, $3$ .

Answer: $3$

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