The Normal Distribution

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The Normal Distribution

• MATH 224

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The Normal Distribution

• Most commonly used distribution in science
• numbers that are combinations of many small
  influences are often normally distributed
• many derived properties are averages, and
  averages have deep connections with normal
  distribution

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PDF Formula
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Interesting Properties

• We can define mean, spread for a distribution
  (not just sets of data)
• Mean of a normal distribution is ______
• Standard deviation of a normal distribution
  is_______

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Standard Normal

• As a reference, XN(0, 1) is a useful reference
  distribution

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In MATLAB
x linspace(-4, 4, 1000) y normpdf(x, 0,
1) plot(x, y)
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Graph Comments
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Interpretation

• What does the value
• normpdf(0, 0, 1) 0.3989
• represent?

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Difficulties

• Normal distribution has many great properties
• Biggest analytic challenge is that
• PDF well defined, but
• CDF probabilities cannot be computed exactly

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Overcome By Numerical Methods

• In the old days, needed a table with normal
  distribution values
• Now can use built-in cdf functions

normcdf(-1.5, 0, 1) normcdf(3, 0, 1)
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Interpretation

• normpdf(0, 0, 1) 0.3989 means
• What does the value
• normcdf(0, 0, 1) 0.5000
• represent?

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Question

• For the standard normal distribution, what is the
  probability that a random value will lie within
  /- 2 standard deviations of the mean?
• normcdf(2, 0, 1)
• normcdf(1, 0, 2)
• normcdf(-2, 0, 1) normcdf(2, 0, 1)
• normcdf(2, 0, 1) normcdf(-2, 0, 1)

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Quick Reference

• 2 middle std dev_________
• 3 middle std dev_________
• 4 middle std dev_________
• larger than 2 std. dev _________

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Reading

• Section 4.3