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  1. MHB Craftsman

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    #1
    Let $$X_1,\dots,X_n$$ be i.i.d $$N(\mu,\sigma^2)$$. What is the marginal pdf of $$\sum_{i=1}^n (X_i-\overline{X})^2$$.

    I'm guessing it's some sort of chi square distribution but how to find this I am unsure. Thanks

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    #2
    Quote Originally Posted by Fermat View Post
    Let $$X_1,\dots,X_n$$ be i.i.d $$N(\mu,\sigma^2)$$. What is the marginal pdf of $$\sum_{i=1}^n (X_i-\overline{X})^2$$.

    I'm guessing it's some sort of chi square distribution but how to find this I am unsure. Thanks
    The chi-square distribution is defined as $ \displaystyle \chi_n^2 = \sum_{i=1}^n Y_i^2$ where $Y_i \sim N(0,1)$.
    We need to rewrite the expression to standard normal distributions to relate it to the chi-square distribution.
    The first step has already been done, since $X_i-\overline X \sim N(0,\sigma^2)$.

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