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If $Y = \sqrt{\sum_{i=1}^N X_i^2} $, where $X_i \sim \mathcal{N}(\mu,\sigma^2)$, i.e. all $X_i$ are i.i.d gaussian random variables of same mean and variance, then what is the resultant PDF of $Y$? How to calculate the mean and variance of the resultant pdf?

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