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This is about notations used in the discussion of Random Variables in the book Schaum's Outline of Probability, Random Variables, and Random Processes by Hwei Hsu, Chapter-2, Page-38.

Let, we flip a coin thrice, and a random variable $X$ represents the number of heads.

Then,

a. $S = \{TTT, HTT, THT, TTH, HHT, HTH, THH, HHH\}$.
b. $\zeta = TTT, HTT, THT, TTH, HHT, HTH, THH, HHH$.
c. $X(\zeta) = x = 0, 1, 2, 3$.
d. $X(\zeta_1) = X(\{TTT\}) = 0 = x_1$
e. $X(\zeta_2) = X(\{HTT\}) = 1 = x_2$
f. $X(\zeta_3) = X(\{THT\}) = 1 = x_2$
g. $X(\zeta_4) = X(\{TTH\}) = 1 = x_2$
h. $X(\zeta_5) = X(\{HHT\}) = 2 = x_3$
i. $X(\zeta_6) = X(\{HTH\}) = 2 = x_3$
j. $X(\zeta_7) = X(\{THH\}) = 2 = x_3$
k. $X(\zeta_8) = X(\{HHH\}) = 3 = x_4$
l. $(X=x) = \{TTT\}, \{HTT, THT, TTH\}, \{HHT, HTH, THH\}, \{HHH\}$.
m. $(X=0)=\{TTT\}$
n. $(X=1)=\{HTT, THT, TTH\}$
o. $(X=2)=\{HHT, HTH, THH\}$
p. $(X=3)=\{HHH\}$
q. $(X\le3) = \{TTT\}, \{HTT, THT, TTH\}, \{HHT, HTH, THH\}, \{HHH\}$.

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Is any of these assumptions incorrect?

Note. kindly point out if there is any typo.

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  • $\begingroup$ $X(\zeta)$ represents the number of heads in the event $\zeta$. Is that what your question is? $\endgroup$ – John Doe Jun 26 '18 at 14:39
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The random variable $X$ is a function $S \to \mathbb{R}$.

Thus, $X(\zeta)$ is the image of $\zeta$ induced by $X$.

That is, $$\begin{align} X(\zeta)&=\{X(z): z \in \zeta\} \\ &= \{X(TTT), X(HTT), X(THT), X(TTH), X(HHT), X(HTH), X(THH), X(HHH)\} \\ &= \{0, 1, 2, 3\}\text{.} \end{align}$$

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  • $\begingroup$ Is this correct: c. $X(\zeta) = x = 0, 1, 2, 3$. ? $\endgroup$ – user366312 Jun 26 '18 at 14:56

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