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:
Heads
0
1
2
3
4
Prob.
1/16
4/16
6/16
4/16
1/16
 Discrete probability models are shown in tables and all
the probabilities exist only at the whole numbers – in
other words P(3.5) = 0.
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
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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.