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Math Help - variance question

  1. #1
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    variance question

    The question askes... to show E[Var (Y|X)] <= Var (Y)
    I think we should apply the formula for E[Var(Y|X)] first...but I am stuck on it...
    Anyone knows wha's the next step? Thanks.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by cm917 View Post
    The question askes... to show E[Var (Y|X)] <= Var (Y)
    I think we should apply the formula for E[Var(Y|X)] first...but I am stuck on it...
    Anyone knows wha's the next step? Thanks.
    Note that by the law of total variance,

    \text{Var}(Y)=E\left[\text{Var}(Y|X)\right]+\text{Var}(E\left[Y|X\right])\implies E\left[\text{Var}(Y|X)\right]=\text{Var}(Y)-\text{Var}\left(E\left[Y|X\right]\right).

    If \text{Var}\left(E\left[Y|X\right]\right)=0, we see that \text{Var}(Y)-\text{Var}\left(E\left[Y|X\right]\right)=\text{Var}(Y).

    If \text{Var}\left(E\left[Y|X\right]\right)>0, we see that \text{Var}(Y)-\text{Var}\left(E\left[Y|X\right]\right)<\text{Var}(Y).

    Therefore, we can say that E\left[\text{Var}\left(Y|X\right)\right]=\text{Var}(Y)-\text{Var}\left(E\left[Y|X\right]\right)\leq\text{Var}(Y).

    Does this make sense?
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