Section 8.4

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Section 8.4 Continuous Probability Models

• Special Topics

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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 0 1 2 3 4
Prob. 1/16 4/16 6/16 4/16 1/16
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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
  dont always consist of whole number values.
• Continuous data must be measured.
• Since a value in a continuous data set isnt
  always a whole number, you cant represent the
  model with a table. Instead, you use a geometric
  area model.

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Theory behind Density Curves

• Lets 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!)

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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.

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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.

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Uniform Density Curve

• Caution You cant 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

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Irregular Density Curve

• An irregular density curve is any geometric shape
  used for probability which isnt 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.

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Example

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

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Homework

• Worksheet 8.4 day 1.