1. Probability density functions
So far, we have talked about probabilities of discrete quantities, such as die rolls and number of bus arrivals, but what about continuous quantities?
2. Continuous variables
A continuous quantity can take on any value, not just discrete ones. For example, the speed of a train can be 45-point-76 km/h.
3. Michelson's speed of light experiment
Continuous variables also have probability distributions. Let's consider an example. In 1879, Albert Michelson performed 100 measurements of the speed of light in air.
4. Michelson's speed of light experiment
Each measurement has some error in it; conditions, such as temperature, humidity, alignment of his optics, et cetera, change from experiment to experiment. As a result, any fractional value of the measured speed of light is possible, so it is apt to describe the results with a continuous probability distribution. In looking at Michelson's numbers, shown here in units of megameters/s, or 1000s of kilometers/s, we see this is indeed the case.
What probability distribution describes these data? I posit that these data follow the famous Normal distribution. To understand what the Normal distribution is,
5. Probability density function (PDF)
let's consider its probability density function, or PDF. This is the continuous analog to the probability mass function, the PMF. It describes the chances of observing a value of a continuous variable.
The probability of observing a single value of the speed of light does not make sense, because there is an infinity of numbers,
6. Normal PDF
say between 299-point-6 and 300-point-0 megameters per second. Instead, areas under the PDF give probabilities. So, the probability of measuring that the speed of light is greater than
7. Normal PDF
300,000 km/s is an area under the normal curve. Parametrizing the PDF based on Michelson's experiments, this is about a 3% chance, since the pink region is about 3% of the total area under the PDF.
To do this calculation, we were really just looking at the cumulative distribution function,
8. Normal CDF
or CDF, of the Normal distribution. Here is the CDF of the Normal distribution. Remember that the CDF gives the probability the measured speed of light will be less than the value on the x-axis. So, reading off the value at 300,000 km/s,
9. Normal CDF
we see that there is a 97% chance that a speed of light measurement is less than that. So, there is about a 3% change it is greater.
We will study the Normal distribution in more depth in the coming exercises, but for right now,
10. Let's practice!
let's review some of the concepts we've learned about continuous distribution functions.