Types of Discrete Random Variables

There are various types of Discrete Random Variables, some of which are as follows:

• Binomial Random Variable
• Geometric Random Variable
• Bernoulli Random Variable
• Poisson Random Variable

Binomial Random Variable

A binomial random variable is a type of discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success, denoted by p. It is named after the Swiss mathematician Jacob Bernoulli.

For example, the number of heads obtained when flipping a coin n times, or the number of defective items in a batch of n items can be modelled using a binomial distribution.

The probability mass function (PMF) of a binomial random variable X with parameters n and p is given by:

[Tex]\bold{P(X=k) = \binom{n}{k} \times p^k \times (1-p)^{n-k}}[/Tex]

where k is a non-negative integer representing the number of successes in n trials.

The mean (expected value) and variance of a binomial random variable are given by:

E(X) = np 

and

Var(X) = np(1-p)

Geometric Random Variable

A geometric random variable is a type of discrete probability distribution that models the number of trials required to obtain the first success in a sequence of independent Bernoulli trials, where each trial has a probability of success p and a probability of failure q = 1 – p.

For example, suppose you are flipping a fair coin until you get ahead. The number of times you flip the coin before obtaining a head is a geometric random variable. In this case, p = 0.5 because the probability of getting a head on any given flip is 0.5.

The probability mass function (PMF) of a geometric random variable is given by:

P(X=k) = (1-p)^k-1× p

where X is the number of trials required to obtain the first success, and k is a positive integer representing the trial number.

The mean (expected value) and variance of a geometric random variable are given by:

E(X) = 1/p

and

Var(X) = (1-p)/p²

Bernoulli Random Variable

A Bernoulli random variable is a type of discrete probability distribution that models a single trial of an experiment with two possible outcomes: success with probability p and failure with probability q=1-p. It is named after the Swiss mathematician Jacob Bernoulli.

For example, flipping a coin and getting a head can be modelled as a Bernoulli random variable with p=0.5. Another example is the probability of a student passing a test, where the possible outcomes are passing with probability p and failing with probability q=1-p.

The probability mass function (PMF) of a Bernoulli random variable is given by:

[Tex]\bold{P(X = x) = \begin{cases} p &, x =1\\ 1 – p &, x = 0 \end{cases}}[/Tex]

Where X is the random variable and 1 represents success and 0 represents failure.

The mean (expected value) and variance of a Bernoulli random variable are given by:

E(X) = p 

and

Var(X) = p(1-p)

Poisson Random Variable

A Poisson random variable is a type of discrete probability distribution that models the number of occurrences of a rare event in a fixed interval of time or space. It is named after the French mathematician Siméon Denis Poisson.

For example, the number of phone calls received by a customer service centre in a given hour, the number of cars passing through a highway in a minute, or the number of typos in a book page can be modelled using a Poisson distribution.

The probability mass function (PMF) of a Poisson random variable X with parameter λ, where λ is the average number of occurrences per interval, is given by:

[Tex]\bold{P(X=k) = e^{-\lambda} \times \frac{\lambda^k}{k!}}[/Tex]

where k is a non-negative integer representing the number of occurrences in the interval.

The mean (expected value) and variance of a Poisson random variable are both equal to λ:

E(X) = λ 

and

Var(X) = λ

Discrete Random Variables are an essential concept in probability theory and statistics. Discrete Random Variables play a crucial role in modelling real-world phenomena, from the number of customers who visit a store each day to the number of defective items in a production line. Understanding discrete random variables is essential for making informed decisions in various fields, such as finance, engineering, and healthcare. In this article, we’ll delve into the fundamentals of discrete random variables, including their definition, probability mass function, expected value, and variance. By the end of this article, you’ll have a solid understanding of discrete random variables and how to use them to make better decisions.