# Summation notation

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.

• Yes, that's sloppy notation. Dec 22, 2020 at 17:25
• @abstrusiosity with a little context, I guess you can convert it to an answer. Dec 22, 2020 at 18:04
• @gunes Ok, I made it an answer. Dec 22, 2020 at 18:22
• 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.
– whuber
Dec 22, 2020 at 18:30

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.