MHB Marginal PDF for Sum of Squares: Identifying Distribution and Calculation Method

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The marginal probability density function (pdf) of the sum of squares $$\sum_{i=1}^n (X_i-\overline{X})^2$$ for i.i.d. normal variables $$X_i \sim N(\mu, \sigma^2)$$ is related to the chi-square distribution. Specifically, the expression can be rewritten in terms of standard normal variables, leading to the conclusion that it follows a chi-square distribution. The transformation shows that $$X_i - \overline{X}$$ is normally distributed with mean zero and variance $$\sigma^2$$. This establishes the connection to the chi-square distribution, confirming that the sum of squares of standardized normal variables yields a chi-square distribution. Understanding these relationships is crucial for accurate statistical analysis and inference.
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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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Fermat said:
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 $$\chi_n^2 = \sum_{i=1}^n Y_i^2$$ 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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