AP Statistics Audio Lectures
Random Variables
by Arnold Kling

"Random Variable" is a mathematical representation of a random process. Analogous to a circle or a triangle geometry

Random variables assign numbers and probabilities to outcomes of random processes; examples:

• the sum of two dice, and the probability of getting each particular sum
XP(X)
21/36
32/36
43/36
54/36
65/36
...
• pick a card at random from a deck, and assign a value of 4 points to an ace, 3 points to a king, 2 points to a queen and 1 point from a jack.
XP(X)
41/13
31/13
21/13
11/13
09/13
• pick a family at random from zip code 20902, and count the number of children
XP(X)
100
9.005
8.012
7.015
6.027
...

Random Variables can be Continuous

• rainfall in October in Silver Spring
• height in centimeters of a pine tree in a forest

think of X as the midpoint of a range, and p(X) as the probability that the random variable will fall within that range

Random Variable Represented as a Histogram

Random Variable and Gambling

A lottery that costs \$1, has 10,000 entrants, a grand prize of \$5000 and four smaller prizes of \$100 each

XP(X)
+ \$49991/10,000
+ \$994/10,000
- \$19,995/10,000

Many mathematical models of random variables (many shapes in geometry)

Two used in this course: binomial; normal (bell curve)

Two others mentioned briefly: geometric; uniform (rectangular)

### Remember

Random variables are mathematical representations of random processes.

We assign a number to each outcome and associate a probability with that number.