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.

X is a number whose values are defined on the outcomes of a random experiment.
∴ X is a random variable.
Now S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Then X(HHH) = Rs. (2 × 3) = Rs. 6 X(HHT) = X(HTH) = X(THH)
= Rs. (2 × 2 - 1 × 1.50) = Rs. 2.50
X(HTT) = X(THT) = (TTH) = Rs. (1 × 2 - 2 × 1.50)
= - Re 1
and X(TTT) = - Rs. (3 × 1.50) = - Rs. 4.50
where, minus sign shows the loss to the player. Thus, for each element of the sample space, X takes a unique value, hence, X is a function on the sample space whose range is {- 1, 2.50, - 4.50, 6}.