# Statistics — Distributions of Discrete random variables

In this article, we would like to introduce three main kinds of distributions of discrete random variables, these are the Bernoulli distribution, the Binomial distribution and the Poisson distribution. We will give each definitions, examples and also the basic properties of the distributions.

**1. Bernoulli Distribution**

1.1. **Definition**

A **Bernoulli distribution** models a single trail of flipping a ‘fair’ coin in essence. It is the probability distribution of a random variable X which is defined on a random experiment that can have two outcomes, ‘0’ or ‘1’. The probability density function of X is shown below:

We could also define our probability density function of X in a single mathematical expression:

1.2. **Examples**

- Tossing a “fair” coin and defining: X=1 if Tails, X=0 if Heads, then X follows a Bernoulli distribution with the probability of success equals 0.5.
- A student takes an exam with the probability 80% to pass: X=1 if passes, X=0 if fails, then X follows a Bernoulli distribution with the probability of success equals 0.8.
- A visitor of a website clicks on an ad with probability 2%: X=1 if clicks, X=0 if not, then X follows a Bernoulli distribution with the probability of success equals 0.02.

1.3. **Basic Properties**

2. **Binomial Distribution**

2.1. **Definition**

In order to introduce **Binomial distribution**, suppose we repeat a Bernoulli trail *n* times, instead of being interested in the entire sequence of outcomes, we would like to know ‘out of the *n* trails, how many resulted in a success’. If we defined X as the number of successes in *n* trails then X follows a Binomial distribution. The probability density function is:

2.2. **Example**

In tossing a ‘fair’ coin 10 times: what is the probability of obtaining 3 heads?

Note that the ten tosses are mutually independent events, the probability assigned to each toss to any outcome equals 0.5. Hence the result is:

2.3. **Basic Properties**

3. **Poisson Distribution**

3.1. **Definition**

The **Poisson distribution** is used to model the number of occurrences of an event within a given period. For instances, the problems such like how many customers will walk into the grocery store within next 30 minutes? How many airplanes will request landing permission within the next hour? Given the average number of times the event occurs over that period, the Poisson distribution could deal with these problems.

However, for the Poisson distribution to be applicable, several conditions has to be hold:

The probability density function of the Poisson distribution is:

Actually, the formula of the Poisson distribution comes from the Binomial distribution. The mathematic solution are shown below:

3.2. **Example**

One company records the number of vistors that visit their website, they find out that an average of 8 vistors access to their website per hour. Assuming the number of vistors that visit their website follows the Poisson distribution, what is the probability that over 10 vistors access to their website per hour?

Therefore, the probability that over 10 vistors access to the company website per hour is 0.2834.

3.3. **Basic Properties**

**Reference**

Brilliant.org (2020) ‘*Possion Distribution*’. https://brilliant.org/wiki/poisson-distribution/

Emvalomatis, G. (2020) ‘*Random Variables and Distributions*’ [PowerPoint presentation]. *BU52018: Business Analytics*.

Khan Academy (2009) ‘Poisson process 2’. https://www.youtube.com/watch?v=Jkr4FSrNEVY