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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 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). 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.
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…and so on. 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.
- 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
Continuous probability distributions are expressed with a formula (a Probability Density Function) describing the shape of the distribution.