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Forecast from a jointly normal pair

StatisticsEasy

$X$ and $Y$ are jointly normal with means $10$ and $20$ , standard deviations $2$ and $3$ , and correlation $0.5$ . You observe $X = 14$ . What is $E [Y ∣ X = 14]$ ? Give a number.

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Hint

The slope of the conditional mean is $ρ$ times the ratio of standard deviations.

Solution

The slope is $0.5 \times \frac{3}{2} = 0.75$ .
The observation sits $14 - 10 = 4$ above its mean.
So $E [Y ∣ X = 14] = 20 + 0.75 \times 4 = 23$ .

Answer: $23$

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