Discrete random variables are those variables who can attain countable, discrete values such as $1,\ 2,\ 3,\ 4$.. The probability distribution of discrete random variables gives the probability of getting each value of the random variable. For example, if a discrete random variable can attain values $1,\ 2,$ and $3$, then the probability distribution will give the probabilities of getting the values $1,\ 2$ and $3$ for $X$. The cumulative distribution will give the probability of $X$ getting a values less than a certain value. 

Word Problems

Let us see some examples to understand the concept of discrete random variables.
Problem 1:

Three coins are tossed simultaneously. Find the value of the random variable for the number of tails obtained.


The outcome of a tossing of a coin is head $(H)$ or tail $(T)$.

The sample space of the random experiment will be $\{HHH,\ HHT,\ HTH,\ THH,\ TTH,\ THT,\ HTT,\ TTT\}$

The number of tails obtained, $X$ can be $\{0,\ 1,\ 2,\ 3\}$.

It can be shown as,

 0  HHH
 3  TTT
Problem 2: 

Find the cumulative distribution of the probability of getting less than $2$ heads when $3$ coins are tossed simultaneously.


The sample space of tossing three coins = $\{HHH,\ HHT,\ HTH,\ THH,\ TTH,\ THT,\ HTT,\ TTT\}$

The cumulative distribution for the random variable representing number of heads is given here,

 $P(X\ \leq\ k)$ 
 0   $\frac{1}{8}$   $\frac{1}{8}$
 1  $\frac{3}{8}$  $\frac{1}{2}$
 2  $\frac{3}{8}$  $\frac{7}{8}$
 3  $\frac{1}{8}$   $1$

Hence, the cumulative distribution for getting less than $2$ heads is $\frac{1}{2}$.
Problem 3:

There are two students who can get $A,\ B,$ or $C$ grade. Find the sample space, and the values for random variable of number of students getting $A$ grade.


The sample space will be, $S$ = $\{AA,\ AB,\ BA,\ BB\}$

Let the random variable showing number of students getting $A$ grade be $X$.

The value of $X$ can be $\{0,\ 1,\ 2\}$

 $X$   Events  $P(X)$ 
 0 BB
 1  AB, BA
 2  AA  $\frac{1}{4}$
Problem 4:

Let there be five boxes having $10$ pens each. A box will be rejected by retailer if it has more than $3$ pens as defected. Find the random variables, and the values they can attain.


There can be two random variables with discrete values.

$X$ = Number of defected pens in a box

The values $X$ can attain are $\{0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10\}$.

Another random variable will be number of defected boxes.

$Y$ = Number of defected boxes

Values $Y$ can attain are $\{0,\ 1,\ 2,\ 3,\ 4,\ 5\}$