Outline

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Outline

• Random Variables
• Discrete Random Variables
• Continuous Random Variables
• Symmetric Distributions
• Normal Distributions
• The Standard Normal Distribution

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1. Random Variables

• Two kinds of random variables
• Discrete (DRV)
• Outcomes have countable values
• Possible values can be listed
• E.g., of people in this room
• Possible values can be listed might be 28 or
  29 or 30

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1. Random Variables

• Two kinds of random variables
• Continuous (CRV)
• Not countable
• Consists of points in an interval
• E.g., time till coffee break

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1. Random Variables

• The form of the probability distribution for a
  CRV is a smooth curve. Such a distribution may
  also be called a
• Frequency Distribution
• Probability Density Function

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1. Random Variables

• In the graph of a CRV, the X axis is whatever
  you are measuring (e.g., exam scores, depression
  scores, of widgets produced per hour).
• The Y axis measures the frequency of scores.

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X
The Y-axis measures frequency. It is usually not
shown.
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2. Symmetric Distributions

• In a symmetric CRV, 50 of the area under the
  curve is in each half of the distribution.
• P(x ?) P(x ?) .5
• Note Because points are infinitely thin, we can
  only measure the probability of intervals of X
  values not of individual X values.

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50 of area
µ
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3. Normal Distributions

• A particularly important set of CRVs have
  probability distributions of a particular shape
  mound-shaped and symmetric. These are normal
  distributions
• Many naturally-occurring variables are normally
  distributed.

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Normal Distributions

• are perfectly symmetrical around their mean, ?.
• have the standard deviation, ?, which measures
  the spread of a distribution an index of
  variability around the mean.

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?
µ
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Standard Normal Distribution

• The area under the curve between ? and some
  value X ? has been calculated for the standard
  normal distribution and is given in the Z table
  (Table IV).
• E.g., for Z 1.62, area .4474
• (Note that for the mean, Z 0.)

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.4474
X
Z 1.62
Z 0
?
Area gives the probability of finding a score
between the mean and X when you make an
observation
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Using the Standard Normal Distribution

• Suppose average height for Canadian women is 160
  cm, with ? 15 cm.
• What is the probability that the next Canadian
  woman we meet is more than 175 cm tall?
• Note that this is a question about a single case
  and that it specifies an interval.

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Using the Standard Normal Distribution
We need this area
Table gives this area
160
175
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µ
Remember that area above the mean, ?, is half
(.5) of the distribution.
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Using the Standard Normal Distribution
Call this shaded area P. We can get P from Table
IV
160
175
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Using the Standard Normal Distribution

• Z X - ? 175-160
• ? 15
• 1.00
• Now, look up Z 1.00 in the table.
• Corresponding area ( probability) is P .3413.

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Using the Standard Normal Distribution
This area is .3413
So this area must be .5 .3413 .1587
160
175
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Using the Standard Normal Distribution
This area is .3413
So this area must be .5 .3413 .1587
Z 0
Z 1.0
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Using the Standard Normal Distribution

• What is the probability that the next Canadian
  woman we meet is more than 175 cm tall?
• Answer .1587

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Review

• Area under curve gives probability of finding X
  in a given interval.
• Area under the curve for Standard Normal
  Distribution is given in Table IV.
• For area under the curve for other
  normally-distributed variables first compute
• Z X - ?
• ?
• Then look up Z in Table IV.