1. Continuous distributions
We can use discrete distributions to model situations that involve count or interval data, but how can we model continuous data?
2. Waiting for the bus
Let's start with an example. The city bus arrives every 12 minutes, so if we show up at a random time, we could wait anywhere from zero minutes if we arrive just as the bus pulls in, up to 12 minutes if we arrive as the bus leaves.
3. Continuous uniform distribution
Let's model this scenario with a probability distribution. There are an infinite number of minutes we could wait since we could wait five minutes, 1.5 minutes, 1.53 minutes, and so on. Therefore, we can't create individual blocks like we could with count or interval data.
4. Continuous uniform distribution
Instead, we'll use a continuous line to represent probability. The line is flat since there's the same probability of waiting any time from zero to 12 minutes. This is called the continuous uniform distribution.
5. Probability still = area
Now that we have our distribution, let's figure out the probability that we'll wait between four and seven minutes. Just like with discrete distributions, we can take the area from four to seven to calculate probability.
6. Probability still = area
The width of this rectangle is seven minus four, which is three. The height is one-twelfth.
7. Probability still = area
Multiplying those together to get area, we get three-twelfths or 25%.
8. Waiting seven minutes or less
What about the probability of waiting seven minutes or less. We can think of this as a space within the total area where the upper limit is seven minutes and the lower limit is zero minutes.
To calculate the probability we subtract the lower limit, in this case, zero out of a possible 12 minutes, from the upper limit, which is seven out of a possible 12 minutes.
Seven minus zero is seven. We divide the result of this by the total area under the line, which is the maximum possible time we might wait, which is 12.
So the probability of waiting seven minutes or less for the bus is seven-twelfths, or approximately 58.3 percent.
9. Total area = 1
To find the probability of waiting anywhere between zero and 12 minutes, we multiply 12 by one-twelfth, which is one,
10. Total area = 1
or 100%. This makes sense since we're certain we'll wait anywhere from zero to 12 minutes.
11. Probability of waiting more than seven minutes
So if we know the total probability is one, and the probability of waiting seven minutes or less is 58.3 percent, then we can use this to find the probability of waiting more than seven minutes.
We simply subtract the probability of waiting seven minutes or less from the total probability. So, one minus seven-twelfths equals five-twelfths, or around 42 percent.
12. Bimodal distribution
Continuous distributions can take forms other than uniform, where some values have a higher probability than others.
For example, here we can see a distribution with two values occurring most frequently. This is known as the bimodal distribution, as there are two modes. An example of the bimodal distribution is book prices, with different typical values occurring depending on whether the book is a paperback or a hardback.
13. The normal distribution
Another continuous distribution has a peak in the middle, with symmetrical slopes as values move away from the center in either direction.
This is often described as a bell-shaped curve and referred to as the normal distribution. It is very common to observe this distribution, such as with blood pressure or retirement age.
We will learn more about this special distribution later in the course.
14. Total area still = 1
Regardless of the shape of the distribution, the area beneath it must alway equal one, as the area covers 100% of possible outcomes.
15. Let's practice!
Now it's time to practice working with continuous distributions!