# Discrete Random Variable

A random variable $X$ is called discrete if it can assume a finite number of values. If two bulbs are selected from a certain lot, the number of defective bulbs may be 0, 1 or 2. The range of the variable is from 0 to 2 and the random variable can take some selected values in this range. The number of defective bulbs cannot be 1.1 or $- 1$ or 3, etc. This random variable can take only the specific values which are 0, 1 and 2. When two dice are rolled, the total on the two dice will be 2, 3, …, 12. The total on the two dice is a discrete random variable.

Discrete Probability Distribution

Suppose a discrete random variable $X$ can assume the values ${x_1},{x_2},{x_3}, \ldots ,{x_n}$ with corresponding probabilities $p\left( {{x_1}} \right),p\left( {{x_2}} \right),p\left( {{x_3}} \right), \ldots ,p\left( {{x_n}} \right)$. The set of ordered pairs $\left[ {{x_1},p\left( {{x_1}} \right)} \right],\left[ {{x_2},p\left( {{x_2}} \right)} \right],\left[ {{x_3},p\left( {{x_3}} \right)} \right], \ldots ,\left[ {{x_n},p\left( {{x_n}} \right)} \right]$ is called the probability distribution or probability function of the random variable $X$. A probability distribution can be presented in a tabular form showing the values of the random variable $X$ and the corresponding probabilities donated by $p\left( {{x_i}} \right)$ or $f\left( {{x_i}} \right)$ . The above information can be collected in the form of a table as below, which is called the probability distribution of the random variable $X$. The probability that the random variable $X$ will take the value ${x_i}$ is denoted by $p\left( {{x_i}} \right)$ where $p\left( {{x_i}} \right) = P\left( {X = {x_i}} \right)$.

 Values of random variable $\left( {{x_i}} \right)$ ${x_1}$ ${x_2}$ ${x_3}$ $\cdots$ ${x_i}$ $\cdots$ ${x_n}$ Probability $p\left( {{x_i}} \right)$ $p\left( {{x_1}} \right)$ $p\left( {{x_2}} \right)$ $p\left( {{x_3}} \right)$ $\cdots$ $p\left( {{x_i}} \right)$ $\cdots$ $p\left( {{x_n}} \right)$

The probability of some interval can be calculated by adding the probabilities of all points in the interval. For example, $P\left[ {{x_1} < X < {x_4}} \right] = P\left( {X = {x_2}} \right) + P\left( {X = {x_3}} \right)$.

This type of addition will not be possible in continuous random variable to find the probability of an interval. Therein we shall use integral calculus to find the probability of an interval.

The probability distribution can also be described in the form of an equation for $f\left( {{x_i}} \right)$ with a list of possible values of the random variable $X$. Some probability distributions in the form of equations are

(i) $f\left( {{x_i}} \right) = \frac{1}{6}$       for       ${x_i} = 1,2,3, \cdots ,6$

For each value of $X$ the probability is 1/6. It is the probability distribution when a fair die is rolled.

(ii) $f\left( {{x_i}} \right) = \left( {\begin{array}{*{20}{c}} 3 \\ x\end{array}} \right){\left( {\frac{1}{2}} \right)^3}$       for       $x = 0,1,2,3$

Example

A digit is selected from the first 8 natural numbers. Write the probability distribution of $X$ where $X$ is the number of factors (divisors) of digits.

Solution

It is assumed here that the probability of selection for each digit is $\frac{1}{8}$. We can write:

 Digit Factors Number of Factors Probability 1 1 1 1/8 2 1, 2 2 1/8 3 1, 3 2 1/8 4 1, 2, 4 3 1/8 5 1, 5 2 1/8 6 1, 2, 3, 6 4 1/8 7 1, 7 2 1/8 8 1, 2, 4, 8 4 1/8

The information in the table above can be summarized as below in the form of a table of probability distribution:

 Number of factors r.v. $\left( {{x_i}} \right)$ 1 2 3 4 Total $p\left( {{x_i}} \right)$ 1/8 4/8 1/8 2/8 1

The probability of $X = 2$ is 4/8 which has been obtained by adding 1/8 four times. Similarly the probability of 4 is 2/8 which has been obtained by adding 1/8 two times.