# Notations regarding Random Variables.

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. $(X\le3) = \{TTT\}, \{HTT, THT, TTH\}, \{HHT, HTH, THH\}, \{HHH\}$.

.

Is any of these assumptions incorrect?

Note. kindly point out if there is any typo.

• $X(\zeta)$ represents the number of heads in the event $\zeta$. Is that what your question is? – John Doe Jun 26 '18 at 14:39

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