Discrete Random Variable

There are two types of random variables: 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 are countable. 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 xP (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. 

The discrete random variable is a variable that can take on 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, where 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) = (xn) 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 = 01 -p     if  x = 1

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^xe^-λ/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; that is, 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. 

• To predict the number of incidents that occur within a given period, we use discrete random variables. 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.