Continuous distribution

< List of probability distributions < Continuous distribution

It’s important to understand the difference between a continuous distribution and a discrete distribution if you’re going to be working with data. These two types of distributions have different characteristics, but they are both important when it comes to making decisions. Let’s take a closer look at what makes them unique.

What is a Continuous Distribution?

A continuous distribution has an infinite range of values. This means that the number of possible outcomes is uncountable, as we often see in variables such as time or temperature. Time is uncountable because it is infinite: you could count from 0 seconds to a trillion seconds…and keep on counting forevermore. Even if you limit time to a range, say between 0 and 2 seconds, it’s still uncountable because of the infinite number of possible divisions, such as 0.2 seconds, 0.0022 second, 0.00004 seconds, and so on. 

A continuous distribution also follows a smooth curve, which is why it is sometimes referred to as a smooth distribution. For example, if you were looking at the temperatures of all the cities in the world over a period of time, the temperature would follow a curve rather than individual points on a graph.

Types of continuous distribution

The normal distribution is the “go to” distribution for many reasons, including that it can be used the approximate the binomial distribution, as well as the hypergeometric distribution and Poisson distribution.
Other continuous distributions that are common in statistics include:

• Beta distribution,
• Cauchy distribution,
• Exponential distribution,
• Gamma distribution,
• Logistic distribution,
• Weibull distribution.
• The Shakil-Singh-Kibria distribution, based on Whittaker functions [2].
• Edgeworth series distribution (ESD), which approximates a probability distribution in terms of cumulants and Hermite polynomials.

What is a Discrete Distribution?

A discrete distribution has a range of values that can be counted. These values may represent integers or fractions, but they are still countable. For example, if you were looking at the number of birthdays cards sent out per year, there would be a finite number of possible outcomes since people can only live up to 122 years old (the age Jeanne Calment lived until). The total range would be from 0 to 122 cards sent out each year.

Other differences

• The probability that a particular random variable will equal a certain value is zero. For example, let’s say you had a continuous probability distribution for men’s heights. What is the probability that a man will have a height of exactly 70 inches? The chart shows that the average man has a height of 70 inches (50% of the area of the curve is to the left of 70, and 50% is to the right). But it’s impossible to figure out the probability of any one man measuring exactly 70 inches. Why not? Imagine measuring a man who is 70 inches tall. It’s unlikely that he’s exactly 70 inches. He’s probably 70.1 inches, or perhaps 69.97 inches. And it doesn’t stop there. He could be 70.1045 inches, or 69.9795589 inches. The fact is, it’s impossible to exactly measure any variable that’s on a continuous scale, and so it’s impossible to figure out the probability of one exact measurement occurring in a continuous probability distribution.
• Continuous probability distributions are expressed with a formula (a probability density function) describing the shape of the distribution. Discrete probability distributions are usually described with a frequency distribution table or other type of graph or chart. For example, the following chart shows the probability of rolling a die. All of the die rolls have an equal chance of being rolled (one out of six, or 1/6). This gives you a discrete probability distribution of:

  Roll	1	2	3	4	5	6
  Odds	1/6	1/6	1/6	1/6	1/6	1/6

References

[1] Kumaraswamy gif: Kelam, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

[2] Shakil, M. et al. (2010). On a family of product distributions based on the Whittaker functions and generalized Pearson differential equation. Pakistan Journal of Statistics 26(1).