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I am reading a statistics book which says:

" If $ X \sim N ( \mu, \sigma^2)$, it is verified that:

$ \sum_{i=1}^{n}X \sim N ( n\mu, n\sigma^2) $

My doubt is if it should have been written as $ \sum_{i=1}^{n}Xi $ or, considering that $X$ is a random variable, this is already implicit.

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    $\begingroup$ Yes, that's sloppy notation. $\endgroup$ Dec 22, 2020 at 17:25
  • $\begingroup$ @abstrusiosity with a little context, I guess you can convert it to an answer. $\endgroup$
    – gunes
    Dec 22, 2020 at 18:04
  • $\begingroup$ @gunes Ok, I made it an answer. $\endgroup$ Dec 22, 2020 at 18:22
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    $\begingroup$ Read literally, the result is false. According to standard mathematical notation, the sum is meaningful and $\sum_{i=1}^n X = nX \sim N(n\mu,n^2\sigma^2).$ The book must have established a context in which it was understood that "$X\sim N(\mu,\sigma^2)$" means there is a sequence $(X_i)$ of iid variables with the given distribution. In that context, the summation can be correctly understood. $\endgroup$
    – whuber
    Dec 22, 2020 at 18:30

1 Answer 1

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You are correct that it should be written as $\sum_{i=1}^{n}X_i$.

Sigma notation requires an index variable and the index variable needs to be indicated in the summand. Sometimes the index is omitted if the context is clear, like $ \sum X $, but if it appears in the Sigma then it needs to be in the $X$, too.

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