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Consider independent random variable $Y_1,Y_2,\dots, Y_n$ and known real numbers $x_1,x_2,\dots x_n$. The distribution of $Y_i$ is Normal with mean zero and variance $\sigma^2/x_i^2$, where $\sigma^2$ is an unknown parameter. Find the MLE of $\sigma^2$?

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$$ L(\sigma) = \prod_{i=1}^n \frac{x_i}\sigma \exp\left( \frac{-x_i^2 Y_i} {2\sigma^2} \right) $$ $$ \ell(\sigma) = \log L(\sigma) = \log x_1+\cdots+\log x_n - n\log\sigma - \frac{x_1^2 Y_i + \cdots+x_n^2 Y_n}{2\sigma^2} $$ $$ \ell\,'(\sigma) = -\frac n \sigma + \frac{x_1^2 Y_i + \cdots+x_n^2 Y_n}{\sigma^3} = \frac{-n\sigma^2 + x_1^2 Y_i + \cdots+x_n^2 Y_n}{\sigma^3} $$ $$ = \left( \frac{x_1^2 Y_i + \cdots+x_n^2 Y_n} n - \sigma^2 \right) \times \text{a positive number}. $$

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