Discrete Random Variable
A discrete random variable is a variable that can take any whole number values as outcomes of a random experiment. The discrete random variable takes a countable number of possible outcomes and it can be counted as 0, 1, 2, 3, 4, ....... Probability distributions are used to show the values of discrete random variables. A discrete random variable is also known as a stochastic variable. Examples of a discrete random variable are a binomial random variable and a Poisson random variable.
Generally, the data can be of two types, discrete and continuous, and here we have considered a discrete random variable. Also, a discrete random variable should not be confused with an algebraic variable. An algebraic variable takes only one value, but a discrete random variable takes numerous values.
What Is Discrete Random Variable?
A discrete random variable is used to quantify the outcome of a random experiment. Discrete Random Variable takes a countable number of possible outcomes. Generally, the discrete random variable can be counted as 0, 1, 2, 3, 4, ....... The data can be of two types, discrete and continuous, and here we consider discrete random variables. Probability distributions are used to show how probabilities are distributed over the values of discrete random variables.
Definition
A discrete random variable can be defined as a type of variable whose value depends upon the numerical outcomes of a certain random phenomenon. It is also known as a stochastic variable. Discrete random variables are always whole numbers, which are easily countable. A probability mass function is used to describe the probability distribution of a discrete random variable.
Random Variable Example
Suppose 2 dice are rolled and the random variable, X, is used to represent the sum of the numbers. Then, the smallest value of X will be equal to 2, which is a result of the outcomes 1 + 1 = 2, and the highest value would be 12, which is resulting from the outcomes 6 + 6 = 12. Thus, X could take on any value between 2 to 12 (inclusive). Now if probabilities are attached to each outcome then the probability distribution of X can be determined.
The discrete random variable should not be confused with an algebraic variable. An algebraic variable represents the value of an unknown quantity in an algebraic equation that can be calculated. However, a discrete random variable can have a set of values that could be the resulting outcome of the experiment.
Mean And Variance Of Discrete Random Variable
The mean and variance of a discrete random variable are helpful in having a deeper understanding of discrete random variables.
Mean Of Discrete Random Variable
The average value of a random variable is called the mean of a random variable. The mean is also known as the expected value. It is generally denoted by E[X]. where X is the random variable. The mean or expected value of a random variable can also be defined as the weighted average of all the values of the variable. The formulas for the mean of a random variable are given below:
 Mean of a Discrete Random Variable: E[X] = \(\sum xP(X = x)\). Here P(X = x) is the probability mass function.
Variance Of Discrete Random Variable
The variance of a random variable can be defined as the expected value of the square of the difference of the random variable from the mean. The variance of a random variable is given by Var[X] or \(\sigma ^{2}\). If \(\mu\) is the mean then the formula for the variance is given as follows:
 Variance of a Discrete Random Variable: Var[X] = \(\sum (x\mu )^{2}P(X=x)\)
Discrete Random Variable  Types
A discrete random variable is a variable that can take on a finite number of distinct values. For example, the number of children in a family can be represented using a discrete random variable. A probability distribution is used to determine what values a random variable can take and how often does it take on these values. Some of the discrete random variables associated with different probability distributions are as follows.
Binomial Random Variable
A random variable that represents the number of successes in a binomial experiment is known as a binomial random variable. A binomial experiment has a fixed number of repeated Bernoulli trials and can only have two outcomes, i.e., success or failure. The number of trials is given by n and the success probability is represented by p.
A binomial random variable, X, is written as \(X\sim Bin(n,p)\)
The probability mass function is given as \(P(X = x) = \binom{n}{x}p^{x}(1p)^{nx}\)
Geometric Random Variable
A geometric random variable is a random variable that denotes the number of consecutive failures in a Bernoulli trial until the first success is obtained. The probability of success in a Bernoulli trial is given by p and the probability of failure is 1  p.
A geometric random variable is written as \(X\sim G(p)\)
The probability mass function is P(X = x) = (1  p)^{x  1}p
Bernoulli Random Variable
A Bernoulli random variable is the simplest type of random variable. It can take only two possible values, i.e., 1 to represent a success and 0 to represent a failure.
A Bernoulli random variable is given by \(X\sim Bernoulli(p)\), where p represents the success probability.
Probability mass function: P(X = x) = \(\left\{\begin{matrix} p & if\: x = 1\\ 1  p& if \: x = 0 \end{matrix}\right.\)
Poisson Random Variable
A Poisson random variable is used to show how many times an event will occur within a given time period. These events occur independently and at a constant rate. The parameter of a Poisson distribution is given by \(\lambda\) which is always greater than 0.
A Poisson random variable is represented as \(X\sim Poisson(\lambda )\)
The probability mass function is given by P(X = x) = \(\frac{\lambda ^{x}e^{\lambda }}{x!}\)
Probability Distribution Of Discrete Random Variable
The probability distribution of a discrete random variable lists the probabilities associated with each of the possible outcomes. It is also called the probability function or probability mass function.
The probabilities of a discrete random variable are between 0 and 1. And the sum of the probabilities of a discrete random variables is equal to 1. The probability distribution of a discrete random variable is similar to normal distribution.
Difference Between Discrete Random Variable And Continuous Random Variable
The following are some of the key differences between discrete random variables and continuous random variables.
 A discrete random variable can take on an exact value while the value of a continuous random variable will fall between some particular interval.
 As a consequence, a probability mass function is used to describe a discrete random variable and a probability density function describes a continuous random variable.
 Examples of distributions with discrete random variable are binomial random variable, geometric random variable, Bernoulli random variable, poison random variable. Examples of distributions with continuous random variable are exponential random variable and normal random variable.
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Examples on Discrete Random Variable

Example 1: What is the mean of a discrete random variable on rolling a dice?
Solution: The discrete random variable, X, on rolling dice can take on values from 1 to 6. The probability of occurrence of each value is 1 / 6.
Using the formula of mean we have, E[X] = \(\sum xP(X = x)\)
Mean of Discrete Radom Variable E[X] = 1·(1 / 6) + 2·(1 / 6) + 3·(1 / 6) + 4·(1 / 6) + 5·(1 / 6) + 6·(1 / 6) = 21 / 6 = 7/2 = 3.5
Answer: Therefore the mean of discrete random variable on rolling a dice is 3.5 
Example 2: Express the probability distribution of the random variable of the sum of the outcomes, on rolling two dice?
Solution:
The sample space of the discrete random variable, for the sum of the outcomes on rolling two dice is S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.
Discrete Random Variable 2 3 4 5 6 7 8 9 10 11 12 Probability(x) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
FAQs on Discrete Random Variable
What Is Discrete Random Variable?
A discrete random variable can be counted as 0, 1, 2, 3, 4, ..... and it is also known as a stochastic variable. Discrete random variables are always whole numbers, which are easily countable. A probability mass function is used to describe the probability distribution of a discrete random variable.
What Is The Use Of Discrete Random Variable?
The discrete random variable is used to represent outcomes of random experiments which are distinct and countable.
What Are The Examples Of Discrete Random Variables?
A discrete random variable is used to denote a distinct quantity. For example, the number of defective light bulbs in a box, the number of patients at a clinic, etc., can all be represented by discrete random variables. Binomial, Geometric, Poisson random variables are examples of discrete random variables.
What Is The Mean Of Discrete Random Variable?
The mean of a random variable is the summation of the products of the discrete random variable, and the probability of the discrete random variable. Mean of a Discrete Random Variable: E[X] = \(\sum xP(X = x)\).
What Is The Variance Of Discrete Random Variable?
The variance of a discrete random variable is the summation of the products of the variance of the random variable from the mean and the probability of the random variable. Variance of a Discrete Random Variable: Var[X] = \(\sum (x\mu )^{2}P(X=x)\)
What Is The Difference Between Variable And Discrete Random Variable?
An algebraic variable in an algebraic equation is a quantity whose exact value can be determined. A random variable is a variable that can take on a set of values as the result of the outcome of an event.
What Is the Difference Between Discrete Random Variable And Continuous Random Variable?
The discrete random variable has whole number values as results and the continuous random variable takes decimals as values of the whole number.
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