AP Statistics Lectures

by Arnold Kling

Random Variables

If we spin a dreidel, the four possible outcomes of the random process are {N,G,H,S}. If we are playing dreidel for toothpicks and there are 8 toothpicks in the center, then the value of G is 8, the value of H is 4, the value of N is 0 and the value of S is -1. When we quantify the values this way, we have a random variable. In this case, we have a random variable that can take on the values of -1, 0, 4, or 8.

Our textbook does a nice job on the topic of random variables. See the bottom of page 367 through the middle of page 377.

Basically, a random variable is a number that is derived from a random process. For example, suppose that we measure the rainfall in July in different cities. One city might have 1.5 inches, another city might have 0.8 inches, and so on. The amount of rainfall is a random variable. Here are some more examples of random variables:

- The number of hits Pedro Martinez allows when he pitches in a game
- The number of students in a class with birthdays in January
- The score that you get when you take the SAT test.

Give some other examples of random variables. Think of some examples that are discrete and some examples that are continuous.

For continuous random variables, we talk about the probability distribution function. The probability distribution function pertains to intervals of a continuous random variable.

One important distribution function is the uniform distribution function. If a random variable is uniformly distributed, that means that the probability of landing in a particular interval is equal to the size of that interval divided by the size of the entire distribution.

For example, consider a random variable that is uniformly distributed between 0 and 100. The entire distribution has a width of 100. The probability of landing between 0 and 10 is 10/100, or 0.1

We can construct all sorts of random variables from a given random process. For example, with a roll of one die, we could assign a value of 10 to even numbers and a value of 3 to odd numbers. The random variable would have values of 10 or 3, depending on what comes up.