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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?

kjetil b halvorsen
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    This is only a slight variation of your other [answered question](https://stats.stackexchange.com/questions/353938/relationship-between-the-gamma-distribution-and-non-central-chi-squared-distribu/354044#354044) - since you already know you need to look at the [noncentral chi distribution](https://en.wikipedia.org/wiki/Noncentral_chi_distribution), it is worth looking up this distribution first. Does this clarify your question? – Ben Jul 02 '18 at 10:38
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    It should be possible to adapt the technique from: https://stats.stackexchange.com/questions/317095/expectation-of-square-root-of-sum-of-independent-squared-uniform-random-variable/317475#317475 – kjetil b halvorsen Jul 02 '18 at 10:56
  • @Ben No sir, no clarity – D Satya Ganesh Jul 02 '18 at 12:55
  • With the information @Ben gave you, you can now [look up the answer](https://en.wikipedia.org/wiki/Noncentral_chi_distribution). Is that what you seek, or do you want to know how the solution is derived? – whuber Jul 02 '18 at 13:43

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