Section 8.4 – Continuous Probability Models Special Topics

Types of Probability Distributions When the values for outcomes only take whole number values, the probability model is called discrete. Discrete values can be counted. An example of a discrete probability model is the number of heads which appear when two coins are flipped: Discrete probability models are shown in tables and all the probabilities exist only at the whole numbers – in other words P(3.5) = 0. Heads 1 2 3 4 Prob. 1/16 4/16 6/16

Types of Probability Distributions The other type of probability model is called continuous. Examples of a continuous setting include blood pressure readings, times to run a race, the height of 4th graders. Continuous values cannot be counted, since they don’t always consist of whole number values. Continuous data must be measured. Since a value in a continuous data set isn’t always a whole number, you can’t represent the model with a table. Instead, you use a geometric area model.

Theory behind Density Curves Let’s say you have a random number generator and you program it to generate any number between 0 and one. You can represent this geometrically with a number line that starts at zero and ends at one. So, what mathematicians do is to make the number line into a square which is 1 x 1. That gives an area = 1 (just like probability!)

Theory behind Density Curves Thus, any geometric figure which has an area equal to 1 is called a density curve. When you “cut” up the density curve and calculate the areas of the pieces, you get numbers which are less than 1 (just like probability). You cut up the density curve from bottom to top. We will look at three density curves in this section – a uniform density curve, a normal density curve, and an irregular density curve. We will look at a normal density curve tomorrow.

Uniform Density Curve A uniform density curve is a rectangle. Our model for the random number generator is a square, so it is also a rectangle. Find the areas of the shaded regions.

Uniform Density Curve Caution: You can’t get probabilities for single numbers when the setting is continuous. This is because a single number would be represented with a line, and a line has NO area geometrically. Thus, P(x = .2) = 0

Irregular Density Curve An irregular density curve is any geometric shape used for probability which isn’t a rectangle or bell-shaped. The area of the curve must be equal to 1. The curve can be a triangle, trapezoid, etc., or any combination of geometric shapes. It, too, is cut up from bottom to top.

Example Is this a density curve? Show mathematical proof! Find P(0.6 < x ≤ 0.8) Find P(0 ≤ x ≤ 0.4) Find P(0 ≤ x ≤ 0.2)

Homework Worksheet 8.4 day 1.