In previous sections, you learned about the fundamentals of descriptive statistics. Specifically, you learned about the properties of measures of central tendency and variability and how they describe the characteristics of distributions. Here, we’ll review the various distributions you will encounter in the probability theory as well as their properties.

Random variables in statistics are variables that can take on any value based on a probability distribution. As you learned in previous sections of this guide, probability distributions, like the normal distribution, tell you how typical a value is given measures like the mean and standard deviation.

Typical values are another way of saying values that are probable and improbable. The classic illustration of a random variable is the flip of a coin. Let's call the flip of a coin X, where the result, which can be either heads or tails, is what makes our random variable random. Here, the probability that it will be heads is 0.5, or 50%.

There are two types of numbers, as we discussed before: discrete and continuous variables. The differences between the two can be summarized in the table below.

Discrete Continuous
Definition Can take on a finite amount of possibilities Can take on an infinite amount of possibilities
Example Colour: Blue, Red, Yellow Height: 1.69 cm, 1.692 cm, 1.693…..9cm
Characteristics
  • Mutually exclusive categories (can only be one or another, not both)
  • Takes on a countable range
  • Not mutually exclusive, can overlap between any two limits (in the previous example, 1.69 can have three “different” versions)
  • Takes on an uncountable range

 

If a random variable is a discrete variable, it takes on a discrete distribution and, likewise, if a random variable is continuous, it takes on a continuous distribution.

Continuous and Discrete Variables

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Discrete Distributions

As we mentioned, random variables which are discrete take on discrete distributions. Distributions are nothing more than the probability of a random variable’s outcome. Therefore, a discrete distribution is nothing more than a discrete probability distribution. In other words, a discrete probability distribution displays each outcome of a random variable and its probability of occurring.

There are three fundamental discrete distributions: binomial, geometric and Poisson distributions.

 

Binomial

Binomial distributions deal with binomial variables, which are variables that can take on only two values. Think of our example before, with our random variable X of flipping a coin. This X could take on only two values: heads and tails.

We can generalize any random variables with two outcomes as being either 0 or 1, where we assign each outcome either a 0 or a 1. The table below is a list of the most common binomial variables, also called Boolean variables.

Outcome 1 Outcome 2
Heads Tails
Yes No
True False
Success Failure
1 0

 

The binomial distribution can be written in the following format,

 

    \[ X \backsim B(n,p) \]

 

This format is how all distributions are written, were a random variable is said to follow a distribution, signalled here by a “B,” and where n and p are the parameters. The parameter n is the number of observations, or sample size, and p is the probability of either outcome 1 or outcome 2 occurring

Naturally, if p was equal to the probability of outcome 1, then the probability for outcome 2 would be,

 

    \[ q = 1 - p \]

 

The image below shows binomial distributions for several n and p.

Binomial Distribution

Geometric

Geometric distributions deal with geometric sequences, which is where each number in the sequence can be found by multiplying the previous number by some fixed constant. This is because the geometric distribution is a probability distribution that indicates the probability of a certain amount of failures before the first success.

The table below illustrates the assumptions that need to be met about a variable in order to use a geometric distribution.

Assumption Example
The outcome of the random variable being studied is found after a series of independent trials The random variable being studied is people’s favourite ice cream flavour at a local ice cream shop
There are only two outcomes for the random variable, either a failure or a success There are only two flavours, vanilla and chocolate, where we deem vanilla a “failure” and chocolate a “success”
The probability for a success is the same in every independent trial The probability of someone liking vanilla is the same for every independent trial because one person preferring vanilla has no effect on whether the next person you ask likes vanilla

The geometric distribution is written as,

 

    \[ X \backsim geom(p) \]

 

Below, you’ll find several geometric distributions with different p.

Geometric Distribution

Poisson

Poisson distributions deal with the outcome of a random variable within a specific time frame, where it depicts the probability for each number of times an outcome can occur within that time interval. Below, you can find the assumptions for a Poisson distribution.

Assumption Example
The number of times an event occurs can take on only integer values k, the number of times an outcome occurs, can only = 0, 1, 5, 100, etc.

For example, the number of times someone calls customer service

The outcomes are independent One person choosing to call customer service does not affect another person choosing to call customer service
Outcomes cannot happen at the exact same time Think about it - even if you place a call at the same time as another customer, there will be milliseconds of difference between the time they connect to customer service

Poisson distributions are written as,

 

    \[ X \backsim Pois(\mu) \]

 

Where the mean, \mu, is the average number of events occurring in the interval given by,

 

    \[ \mu = \lambda t \]

 

Below are Poisson distributions with different \lamda

Poisson Distribution

 

Discrete Distributions

Binomial Geometric

Poisson

Deals with Boolean variables: ones that only take on two values Variables in a geometric sequence, where the probability is of the first “success” Indicates the probability of an event occurring a certain amount of times in a given time frame.
Example:

  • Flipping a coin, can only be either heads or tails
Example:

  • Polling people at an ice cream stand about favourite flavours until the first chocolate is found
Example:

  • How many people call customer service within an hour interval

    \[ X \backsim B(n,p) \]

    \[ X \backsim geom(p) \]

    \[ X \backsim Pois(\mu) \]

 

Continuous Distributions

As the name suggests, continuous distributions display the probability of continuous random variables. These can be variables like heights, test scores, speed, etc. While there are a couple of basic distributions, the most important one to know is the normal or standard normal distribution.

 

Normal

As we’ve showed you before, the normal distribution models normal variables, which follow the 68-95-99.7 rule. The properties for a normal distribution are listed in the table below.

Property Explanation
Shape A normal probability distribution is shaped like a bell curve, which is why it’s often called a “bell curve”
Distribution Its distribution is symmetric around the mean, which means the two sides before and after the mean mirror each other
Rule 68% of the data falls within 1 \sigma of the mean

95% of the data falls within 2 \sigma

99.7% fall within 3 \sigma

It is written as,

 

    \[ X \backsim N(\mu, \sigma) \]

 

Below are normal distributions with different s.

Normal Distributions

 

Standard Normal

A standard normal distribution is the same as a normal distribution. The only difference is that the data points are standardized using the standardization, or z-score, formula.

 

    \[ A = \frac{X-\mu}{\sigma} \]

 

It is written as,

 

    \[ X \backsim N(0, 1) \]

 

Below is a standard normal distribution. Notice that the mean is always at 0.

Standard Normal Distribution

 

Continuous Distributions

Normal

Standard Normal

Models continuous variables with a normal distribution

Models continuous variables that have been standardized

    \[ X \backsim N(\mu, \sigma) \]

    \[ X \backsim N(0, 1) \]

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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.