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Let me denote a standard normal random variable by $X$ and its outcome (sample) by $x$.

My question is about the $\sim$ notation.

Should we write \begin{equation} X \sim \mathcal{N}(0, 1) \end{equation}

or ,

\begin{equation} x \sim \mathcal{N}(0, 1). \end{equation}

i.e. Does the $\sim$ denote a random variable or a sample?

Assume that I have $M$ samples $\{x_1, \dots, x_M \}$ of $X$, how do you denote that using an expression in terms of $x_k$? \begin{equation} x_k \sim \mathcal{N}(0, 1), \quad k = 1, \dots, M. ? \end{equation}

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3 Answers 3

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The conventional notation is to use upper case letters for random variables and lower case letters for numbers (values of the random variables). When you plan to take a sample of size $m$, you denote this sample by upper case letters $X_1,\dots,X_m$ just because the results are not known yet, and $X_k \sim N(0,1)$ for each $k$, and the $X_k$'s are independent random variables. After the sample is taken, you have numbers $x_1,\dots,x_m$ which are denoted by lower case letters. It is not correct to say $x_k \sim N(0,1)$ because $x_k$ is a number and does not have a distribution. $X_k$, on the other hand, does.

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    $\begingroup$ It is not popular to use lowercase for random variables but it is not prohibited and it was and is often used like this. Notation in mathematics is not that strict as it appears. $\endgroup$
    – Tim
    Mar 3, 2017 at 20:31
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The "~" denotes the distribution of a certain random variable, X in your case. To denote the samples, you can use x as a realization of X, or use directly {x1,...,xM}.

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The notation

$$ X \sim \mathcal{f} $$

means that $X$ follows distribution $f$. If it follows some distribution, it must be a random variable. For random variables we usually use uppercase letters and for their realizations we use lowercase. However anyone is free to use any notation he wants and nothing prohibits you for using lowercase letters for random variables.

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  • $\begingroup$ I fully agree with this excellent point. However, the question specifically states $X$ represents an rv and $x$ an outcome, so I don't think it's asking about conventions concerning upper and lower case usage. $\endgroup$
    – whuber
    Mar 3, 2017 at 23:38

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