Probability densities: Curves and continuous random variables

- When we have discrete distributions, they tend to look like this. We can use bar charts because each value is discrete. These are really the only possible outcomes, but when we have continuous random variables, there are an infinite number of possible outcomes. For example, if we spin this wheel, where might the arrow land? It could land on 0.2575, or at 0.5245, or even at 0.9887.

As you can see, the possible outcomes are infinite. Bar charts won't work here, so instead, we use curves to illustrate the distribution of outcomes. These curves are called probability densities. The area under the curve represents the probability of each and every outcome. For this probability density, the probability of outcome A is X. The probability of outcome B is only Y.

Since the area under the curve represents all possible outcomes, then the area under the curve is equal to 1.0 or 100%. Because the probability of any exact single scenario like A or B is so low, most of the time, people will calculate the probability of a group of outcomes. Let's go back to our spinning wheel. We might ask, what is the probability the arrow will land on or between 0.25 and 0.50? If this was the probability density for the spinning wheel, we could see that the area under the curve between points 0.25 and 0.50 would represent the probability that the arrow would land between those two points.

If we wanted to know the probability that the arrow would fall between 0.25 and 0.75, we can now see that the probability is easily over 50% since the shaded area under the curve is a significant percentage of all the area under the curve. Of course, if we ask, what are the odds that the arrow will fall between zero and 0.9999, it would be very close to 100%.

How do you calculate the area under the curve? For that, you will need the formula of the curve, often called the function, and then you'd use calculus. That's probably a bit more than we wanna cover here, but hopefully, you can walk away knowing some simple things. For example, for this probability density, the odds of outcome A are very low, and the probability of outcome B is higher than the probability of outcome A.

The probability of having an outcome between points A and B is certainly less than 25%. A large percentage of the area under the curve is not shaded in. With that in mind, let's now learn about one of the most famous curves of all, the bell-shaped curve.