# About the $X\sim \mathcal{N}$ notation

If a random variable $X$ has mean $\mu_{X}$ and variance $\sigma_{X}^{2}$, and follows a normal distribution, it may be written as

$X\sim \mathcal{N}(\mu_{X},\sigma_{X}^{2})$.

Suppose $Y$ is also normally distributed variables. Should I now write

$X+Y\sim \mathcal{N}(\mu_{X}+\mu_{Y},\sigma_{X}^{2}+\sigma_{Y}^{2})$

or

$(X+Y)\sim \mathcal{N}(\mu_{X}+\mu_{Y},\sigma_{X}^{2}+\sigma_{Y}^{2})$

or something else? Is there some standard notation for this?

• The mean of the new variable would be the average of $\mu_X$ and $\mu_Y$, not their sum. Jul 20 '15 at 18:30
• Maybe I misread, but the only difference seems to be the parenthesis. This (clearly) doesn't matter. Jul 20 '15 at 18:31
• If $X$ and $Y$ are jointly normal, then $X+Y$ is also normal with mean $\mu_X+\mu_Y$ as you state, but the variance of $X+Y$ can be any number ranging from $(\sigma_X-\sigma_Y)^2$ to $(\sigma_X+\sigma_Y)^2$ with the value $\sigma_X^2+\sigma_Y^2$ that you want to use being achieved exactly when $X$ and $Y$ are uncorrelated random variables (including, as a special case, independent random variables). Jul 20 '15 at 20:28
• The $\sim$ notation takes the same precedence as equality. So $x + y \sim \mathcal{N}$ is no more ambiguous than $x + y = z$ and not thinking of $y=z$ as some "operation". They are distinct sides of a certain type of equation ($\sim$ means "has distribution function equal to") Jul 20 '15 at 20:35
• Until you modify your question sufficiently to make the claim true you shouldn't write that in either form but the round brackets/parentheses make no difference notationally (i.e. once fixed by including the additional conditions needed to make the assertion true it's fine to either include or omit the parentheses) Jul 21 '15 at 0:39

You can certainly write $X + Y \sim N(\mu_{X} + \mu_{Y}, \sigma^{2}_{X} + \sigma^{2}_{Y})$ because if $X$ and $Y$ are normally distributed with the parameters you suggest and are independent, then this statement is true.
However, generally you would write something like $Z = X + Y$, therefore $Z \sim N(\mu_{X} + \mu_{Y}, \sigma^{2}_{X} + \sigma^{2}_{Y})$ so that you can start making statements about, and manipulating, $Z$.