Simple example of DRV involving number of heads

Consider tossing a fair coin twice in succession and we are interested in the random variable which represents the number of tails obtained.

Define \(X\) to be the number of tails obtained in two tosses of a fair coin.

In this case, \(X\) can only take on values from the finite set \(\{0,1,2\}\). To each of these values, we have:

\[\mathrm{P}(X = 0) = \frac{1}{4}, \quad \mathrm{P}(X = 1) = \frac{1}{2}, \quad \mathrm{P}(X = 2) = \frac{1}{4}\]

So \(X\) is a random variable whose probability density function is given by

\[\mathrm{f}_X(x) = \begin{cases} \frac{1}{4} & \text{if}\hspace{0.5em} x = 0 \\ \frac{1}{2} & \text{if}\hspace{0.5em} x = 1 \\ \frac{1}{4} & \text{if}\hspace{0.5em} x = 2 \end{cases}\]