Download 1 The Normal (Gaussian) Distribution The Normal (Gaussian

The Normal (Gaussian) Distribution
Section 4.3
1. Most used continuous distribution (as probability model) in statistics
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Also known as Bell Curve
2. Is used for many physical measurements
– heights, weights, test scores (also for errors in measurement)
3. “Central Limit Theorem”
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sums, averages of random variables are often (close to) normally
distributed
4. A two parameter “family” of distributions
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The Normal (Gaussian) Distribution
Notation:
~ short hand for (is distributed as)
Definition (via pdf):
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The Normal (Gaussian) Distribution
Geometry
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The Normal (Gaussian) Distribution
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The Standard Normal Distribution
Special case:  = 0 and 2 = 1 (Standardized Scores)
cdf:
Can not be expressed in
closed functional form!`
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Standard Normal Distribution CDF
Table A3 provides (z) for z = -3.49, -3.48, ..., 3.48, 3.49
(-3.49) = 0.0002 and (3.49) = 1 - (-3.49) = 0.9998
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Standard Normal Distribution CDF
(Table Exercise)
Example: Find P(Z ≤ 1.25) =
Example: Find z such that P(Z ≤ z) = 0.05 (5th percentile)
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Transforming (Standardizing) Normal RV’s
Idea: Transform a N(, 2) RV into a N(0, 1) RV...
then:
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Using the Transformation
Say
and we want to compute
Idea: Transform to the standard normal distribution:
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Using the Transformation: Example
Say the reaction time for a person to respond to brake lights is normally
distributed, with a mean reaction time of 1.25 sec and a standard
deviation of 0.46 sec. What is the probability that the reaction time for a
randomly selected person is between 1.00 sec and 1.75 sec?
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Using the Transformation: Percentiles
Again say X  N(1.25, 2 = (0.46)2 ). What is the 66th percentile of X?
Compute 66th percentile of standard normal distribution and then transform
to get 66th percentile for N(1.25, 2 = (0.46)2) distribution.
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Using the Transformation: Percentiles
Again say X  N(1.25, 2 = (0.46)2 ). What is the 66th percentile of X?
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