$A$ person plays a game of tossing a coin thrice. For each head,he is given $Rs. 2$ by the organiser of the game and for each tail,he has to give $Rs. 1.50$ to the organiser. Let $X$ denote the amount gained or lost by the person. Show that $X$ is a random variable and exhibit it as a function on the sample space of the experiment.

(N/A) random variable is a real-valued function whose domain is the sample space of a random experiment. Since $X$ assigns a unique real number to each outcome of the experiment,$X$ is a random variable.
The sample space $S$ of tossing a coin thrice is:
$S = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$
Let $H$ denote the number of heads and $T$ denote the number of tails. The gain or loss $X$ is given by $X = 2H - 1.50T$.
Calculating $X$ for each outcome:
$X(HHH) = 2(3) - 1.50(0) = 6$
$X(HHT) = 2(2) - 1.50(1) = 4 - 1.50 = 2.50$
$X(HTH) = 2(2) - 1.50(1) = 2.50$
$X(THH) = 2(2) - 1.50(1) = 2.50$
$X(HTT) = 2(1) - 1.50(2) = 2 - 3 = -1$
$X(THT) = 2(1) - 1.50(2) = -1$
$X(TTH) = 2(1) - 1.50(2) = -1$
$X(TTT) = 2(0) - 1.50(3) = -4.50$
Thus,$X$ is a function from $S$ to $\mathbb{R}$ with the range $\{-4.50, -1, 2.50, 6\}$.