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Discrete vs. Continuous Variables.

## What is a Continuous Probability Distribution?

Probability distributions are either **continuous probability distributions **or discrete probability distributions. A continuous distribution has a range of values that are infinite, and therefore uncountable. For example, time is infinite: you could count from 0 seconds to a billion seconds…a trillion seconds…and so on, forever. A discrete distribution has a range of values that are countable. For example, the numbers on birthday cards have a possible range from 0 to 122 (122 is the age of Jeanne Calment the oldest person who ever lived).

## Discrete vs. Continuous Probability Distributions

A **discrete probability distribution** is made up of discrete variables, while a continuous probability distribution is made up of continuous variables. The two types of distributions differ in several other ways.

**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

## Types of Continuous Probability 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.

Less common continuous distributions — ones you’ll rarely encounter in basic statistics courses— include:

- The Shakil-Singh-Kibria distribution, based on the Whittaker functions [1].
- Edgeworth series distribution (ESD), which approximates a probability distribution in terms of cumulants and Hermite polynomials.

## References

[1] 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).

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