On a calm day, with probability 1/2, a return Y has mean 0 and variance 4. On a stormy day, with probability 1/2, it has mean 6 and variance 10. What is Var(Y) for a randomly chosen day?

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Calm and stormy days

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On a calm day, with probability $\frac{1}{2}$ , a return $Y$ has mean $0$ and variance $4$ . On a stormy day, with probability $\frac{1}{2}$ , it has mean $6$ and variance $10$ . What is $Var (Y)$ for a randomly chosen day?

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Hint

Add the average within-regime variance to the variance of the regime means.

Solution

The average within-regime variance is $E [Var (Y ∣ regime)] = \frac{4 + 10}{2} = 7$ .
The regime means $0$ and $6$ are equally likely, so their variance is $3^{2} = 9$ .
Adding the parts gives $Var (Y) = 7 + 9 = 16$ .

Answer: $16$

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