Continuous Probability Distributions

In the next part of the lesson, Sebastian shows you how to visualize and make calculations with a continuous probability distribution. Before you move on, let's compare a discrete and continuous distribution again.

At the very beginning of the lesson, we showed you a discrete probability distribution next to a continuous probability distribution. Here they are again:

The discrete distribution is broken up into slices. Each slice represents an outcome like zero heads, one heads, two heads, or three heads.

The continuous distribution has an un-broken line across the entire x-axis range. You could have a velocity of 20 or 20.5 or -10.451.

Notice the y-axis label on the continuous distribution: "probability density function". For the discrete probability distribution, the y-axis represented the probability of an event occurring. In the continuous case, the probability density function does not represent probability directly; instead, the area underneath the density function curve represents probability.

You'll learn more about this in the next part of the lesson.

But without knowing what "probability density function" even means, you can tell that it's more likely that the velocity is around 20 and less likely that the velocity is around 0 or 40.

Characteristics of a Continuous Distribution

Here are a few characteristics of a continuous distribution and the probability density function. Keep these in mind as you go through the next part of the lesson.

• The y values must be greater than or equal to 0.
• The probability of a specific x value occurring is equal to 0
• The probability of an event occurring between two values of x is equal to the area under the curve between those two x values.
• The total area under the probability density function curve is equal to 1.

In practice, these rules mean that the probability that velocity equals exactly 20 is zero. For a continuous distribution, you can only calculate a probability between a range of values like 19.99 and 20.01.

Because the total area under the curve is 1, there is a 100% chance that the velocity has some value between negative infinity and positive infinity.

Uniform Continuous Distribution

There are many different types of continuous distributions: link to list of continuous probability distributions.

But they all have the same characteristics described above. To calculate probabilities with a continuous distribution, you have to calculate the area underneath a curve. Calculating the area under a curve like in the above visualization requires calculus or a software program.

So Sebastian has chosen to use a specific continuous distribution called the uniform continuous distribution. The uniform continuous distribution forms a rectangle. So you can calculate the area underneath the curve simply by multiplying the base by the height.

Below is an example of a uniform continuous probability distribution. Sebastian will explain more about where this distribution comes from and how to use it.