Law of Total Variance - Quant Trader Interview Question
Difficulty: Medium
Category: Probability & Statistics
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Topics: probability, variance, conditional-expectation
Problem Description
You are analyzing the volatility of a stock. Let $Y$ be the daily return of the stock, and let $X$ be a signal derived from a machine learning model predicting market regime. You want to understand how much of the total variance of the stock's return $Y$ can be explained by the variance within each regime $X$, and the variance between the expected returns of each regime.
Express $\text{Var}(Y)$ in terms of $E\text{Var}(Y|X)$ and $\text{Var}(EY|X)$.
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