11.0 Probability Distribution of Random Variables

Corresponding to every outcome of a random experiment, we can associate a real number. This correspondence between the elements of the sample space associated to a random experiment and the set of real numbers is defined as a random variable.

Let $S$ be the sample space associated with a given random experiment. Then, a real valued function $X$ which assigns to each event $\omega \in S$ to a unique real number $X(\omega )$ is called a random variable.

In simple words, if a random variable assumes countable number of values, it is called a discrete random variable.

Consider a random experiment of tossing three coins.

The sample space of eight possible outcomes of this experiment is given by,

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

Let $X$ be a real valued function on $S$, defined by

$X(\omega )$ = Number of heads in $\omega \in S$.

Then, $X$ is a random variable such that:

$$\begin{equation} \begin{aligned} X(HHH) = 3,X(HHT) = 2,X(HTH) = 2,X(THH) = 2 \\ X(HTT) = 1,X(THT) = 1,X(TTH) = 1,X(TTT) = 0 \\\end{aligned} \end{equation} $$

If $\omega $ denotes the event of "getting two heads" , then $\omega = \{ HTH,THH,HHT\} $ and, $X(\omega ) = 2$.

Similarly, $X$ associates every other compound event to a unique real number.

The range of the random variable $X$ is, $X = \{ 0,1,2,3\} $.

The random variable $X$ can also be described as the number of heads in a single throw of three coins.

Explanation 2: Consider the random experiment of throwing an unbiased die.

Let $Y$ be a real valued function defined on the sample space,

$$S = \{ 1,2,3,4,5,6\} $$

associated with the random experiment, defined as,

$$Y(\omega ) = \left\{ \matrix{1,\;if\;the\;outcome\;is\;an\;even\;number \hspace{1em} \cr- 1,\;if\;the\;outcome\;is\;an\;odd\;number \hspace{1em} \cr} \right.$$

and $\omega $ is an event of occurrence of a number.

Then, by definition,

$$\begin{equation} \begin{aligned} Y(1) = - 1,Y(2) = 1,Y(3) = - 1, \\ Y(4) = 1,Y(5) = - 1,Y(6) = 1 \\\end{aligned} \end{equation} $$

Here the range of $Y$ is, $Y = \{ - 1,1\} $

Therefore, we say that $Y$ is a random variable such that it assumes values $-1$ and $1$.