Continuous random variables

Recall that a discrete random variable is one that can only take on one of a number of discrete values, and nothing in between. For example, a die (singular of "dice") can come up 1, 2, 3, 4, 5 or 6, but not 1.5 or 2.3. There are six discrete choices for the number that shows up on a die, but which of the six turns up on a fair throw is a matter of random chance. That's a discrete random variable.

In this section, we'll focus on continuous random variables (crv). These are variables that can have a continuum of values. Basically that means: you pick two values of the variable, and I can always find another in between. In fact, there are can be an infinite number of values of a crv, but as we'll see, there need not be in order for the kind of thinking about continuous variables that we'll develop to apply.

Dependent & Independent variables

A continuous random variable is one that can take on any value on a continuum of values. That is, if you can pick two values that the variable can take on, I can always find one in between.

Practically, we often used continuous variables and continuous probability distributions to model discrete data, like populations.

Example: Sum of n dice

Let's think about a familiar example, throwing dice. We know that if we toss one die, we're talking about discrete probability. The only values that can come up on a fairly-thrown die are 1, 2, 3, 4, 5 or 6.

But let's expand our experiment to measure the sums of many throws of six dice.

Now there are many more possibilities for that sum, from 6 x 1 = 6 to 6 x 6 = 36. The graph below shows a simulation, done with a spreadsheet, of 500 throws of 6 dice. There are so many possible values of the sum in this case that the distribution of the bars in the graph kind of meld together, and get very close (with some "noise" or unavoidable random fluctuations) to the purple normal distribution in the back.

OK, let's take this example a little bit farther and toss 12 dice 500 times. Now there are many more possible values for the sum (from 12 to 60), and you can see (again with a bit of noise – random fluctuations) that the distribution is very close to a normal distribution.

This trend would continue if we added more dice. Imagine tossing a thousand. The number of possibilities for the sum is very large. The more possibilities, the smaller the distance between them, relative to overall span of the possibilities. Hopefully you can see how the distribution would continue to smooth out and approach that bell-shaped curve model.

A continuous distribution: Heights of 4-year olds

Now let's consider a real continuous distribution. By looking at some census data, I've simulated the distribution of the heights of 4-year-old children (in inches) in the United States. Assuming that medical workers try to measure heights precisely and accurately, these are truly continuous data.

You can see by looking at the graph that the average height is 40 inches, and that there are about as many kids taller and shorter than 40 inches. If you gave me two heights, say 39 and 40 inches, I could find a number of kids around 39-1/2 inches tall, and we could find heights that varied by 1/32 of an inch or so, though of course that's kind of meaningless for heights. In a week a four year old will be 1/32 of an inch taller!

There is much more to continuous probability. We have to have ways to measure the spread of values in a distribution of continuous random variables, and to measure the quality of the data in order to make decisions from it and about it,

and ways to "normalize" distributions so that they can be more easily and meaningfully compared. I'll leave that for another section on probability distributions.

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