From the very beginning I have used the notation $ S = \{ H , T \} $ as the Sample space for tossing a coin once and $ S = \{ HH , HT , TH , TT \} $ in the case of tossing a coin twice.I have several text books using the same method as you can see in A CONCISE COURSE IN A-LEVEL STATISTICS by J CRAWSHAW and J CHAMBERS. Recently in a workshop one of our professors was saying the correct way to use simple letters $ h, t $ instead of capital letters $ H, T $ because capital letters are used to denote events in a Sample space of the random experiment. What would you suggest as the standard correct form to continue?
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4$\begingroup$ Another convention is ... since $H$ and $T$ are not variables, they are constants, use roman type for them: $\{\mathrm{H},\mathrm{T}\}$ not $\{H,T\}$. $\endgroup$– Gerald EdgarCommented May 10 at 13:29
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$\begingroup$ @Gerald Edgar thanks for pointing out that. Though I haven't thought about it may the convention used in many text books. $\endgroup$– Janaka RodrigoCommented May 10 at 23:43
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$\begingroup$ @GeraldEdgar: good point, although the distinction may be lost in handwriting. $\endgroup$– J WCommented May 16 at 14:59
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2 Answers
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It's true that you typically refer to a random variable by a capital letter (e.g., $X$) and a particular value by a lowercase letter (e.g., $x$).
So, $P(X = x)$ would represent the probability that the random variable $X$ takes the value $x.$
This notation is helpful because it helps you keep track of what is a variable and what is a particular value.
In your case, you're talking about a coin flip that reveals a face of the coin, so I would probably call your random variable $F$ and say that it can take on two values, $h$ (heads) or $t$ (tails).
So, for a fair coin, you'd say $P(F = h) = 0.5$ and $P(F = t) = 0.5.$ The sample space over one flip would be $\{ h, t \}$ and over two flips would be $\{ hh, ht, th, tt \}.$
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$\begingroup$ What do you think about the comment of Gerald Edgar ? I know when you come to hand writing it might be not clearly identified. If you can agree with that we can have a better answer including that issue. $\endgroup$ Commented May 10 at 23:50
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1$\begingroup$ @JanakaRodrigo I would agree with Gerald Edgar's comment. $\endgroup$ Commented May 11 at 0:16
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There is no correct here. While it is common to use capital letters for random variables and the corresponding lowercase letters for the values they assume, this convention is not a strong one and often is not followed with regards to simple discrete random variables. It is also common to write the sample space for two coin tosses as $\{HH, HT, TH, TT\}$. For example, in the first chapter of the excellent undergraduate textbook by Grimmett and Stirzaker, the sample space for a single coin toss is written $\{H, T\}$.
When one writes such a simple sample space explicitly in the classroom, normally one has not yet begun to speak of random variables and so conventions regarding random variables are largely irrelevant. In the particular case of coin tosses the use of uppercase letters is also common because of readability issues on the blackboard that usually are more serious when writing lowercase letters - in particular $h$ sometimes can be confused with $n$ and $t$ often can be confused with $+$ if the teacher does not have particularly nice handwriting.
