Discrete Random Variable

Random variables are of two types: discrete and continuous. Discrete random variables take on a countable set of distinct values, such as 0, 1, 2, 3,  …. The outcomes of a random experiment can be counted. The probability distribution of discrete random variables is represented using a probability mass function.

Random variables can be classified into two types: discrete and continuous. Here is the difference between them.

The mean and variance are used to describe the behavior of a discrete random variable. Here, we will discuss what the mean and variance of discrete random variables, denoted E[X], is its expected value.

Mean of Discrete Random Variable: 

The mean of a discrete random variable is the average value of a random variable. It is represented as E[X], where X is the random variable. The mean is also known as the expected value and weighted average. 

The formula to calculate the mean of a discrete random variable is:

E[X] = Σ x × P(X = x)

Where P(X = x) is the probability mass function

Variance of a Discrete Random Variable: 

The average of the squared deviations of the random variable from its mean is the variance of the discrete random variable. It can be represented as Var[X] or σ². 

The variance of discrete random variables is calculated using the formula: 

Var[X] = Σ (x - µ)² × P(X = x)

Here, µ is the mean, and p(X = x) is the probability of each value. 

A discrete random variable can take specific countable values. It represents the possible outcomes of a random experiment and the probability of each outcome. The types of discrete random variables are:   

• Binomial Random Variables

• Geometric Random Variables

• Bernoulli Random Variables

• Poisson Random Variables

The binomial random variables are the possible outcomes of a binomial experiment. A binomial random variable involves a fixed number of independent Bernoulli trials. Each result is either a success or a failure. 
 

• It is represented as X ∼ Bin(n, p), where X is the binomial random variable. 
   
• For a binomial random variable, the probability mass function:
  
  \(P(X = x) = C(n, x) \space p^x (1 - p)^{n - x}\)

The geometric random variables represent the number of trials needed to get the first success in a sequence of independent Bernoulli trials. In a geometric random variable, the probability of success is denoted by p, and the probability of failure is 1 - p

• The geometric variable is represented by: X ∼ Geom(p)
   
• For a geometric random variable, the probability mass function is: 
  
  \(P(X = x) = (1 - p)^{x - 1} × p\)

A Bernoulli random variable represents an experiment with two possible outcomes, i.e., 1 for a success and 0 for a failure.
 

• It is represented by X ∼ Bern(p). 
   
• The probability mass function of a Bernoulli random variable is: 
  
  P(X = x) = { p if x = 1; 1 - p if x = 0 }

A Poisson random variable represents the number of events that occur within a fixed time interval of time or space, where the events occur independently and at a constant rate.
 

• It is denoted by X ∼ λ, where λ is the parameter of a Poisson distribution, which is always greater than 0. 
   
• The probability mass function is:
  
  \(P(X = x) = {(λ^x \times e^-λ) \over x!}, x =0, 1, 2, .....\)

The probability distribution of a discrete random variable is the probability of each of its possible outcomes. When constructing a distribution table, the probability should follow these conditions:

• Each probability value must lie between 0 and 1: 0 ≤ P(X = x) ≤ 1 for each possible outcome.

• The sum of the probabilities of all possible outcomes must be equal to 1

In fields like quality control, statistics, and finance, we use discrete random variables. In this section, we will learn some real-world applications of discrete random variables.

• In quality control, we use discrete random variables to find the number of defects in a batch and also to identify the quality issues. 

• Discrete random variables help predict the number of incidents occurring in a given period. For example, to find the number of crimes reported in a city per day. 

• In call centers, discrete random variables, such as the number of calls per hour, help manage staffing and performance.