The Normal Distribution

1
The Normal Distribution

• MSIT3000
• Lecture 08

2
Objectives

• Be able to calculate probabilities based on the
  Normal Distribution.
• Explain why the Normal Distribution is so
  important.
• Text 4.7

3
The Probability Density Function
4
Notation

• If a variable X is normally distributed with mean
  and standard deviation we write
• XN()
• If X has a mean of zero and a standard deviation
  of one we call X standard normally distributed
  (and we usually use Z instead of X)
• ZN(01)

5
Calculating probabilities

• Recall that P(XA)0 for any specific value of A.
  
• This is true for all continuous probability
  distributions.
• Therefore we can only calculate probabilities of
  intervals
• P(XltA)
• To help us we find the Cumulative Density
  Function F(A) P(XltA)

6
The Standard Normal Cumulative Distribution
7
Comparison of pdf to cdf
8
Calculating probability

• To find the probability that X is less than A
  look up A in the CDF and read off the
  probability.
• We use a table to find a numerical value (see the
  back of the text).
• Problem the CDF is different for every
  difference in or .
• Solution convert to standard normal
• i.e. x to z

9
Converting X to Z

• If X is normally distributed then
• is standard normally distributed.
• Therefore P(XltA)P(Zlt(A-)/) is something we
  can easily look up in the table.

10
Tricks for finding probabilities

• The normal distribution is symmetrical so if
  0 then P(X lt A) P( X gt -A ).
• This is useful because the table does not have
  entries for negative values of A.
• P(X gt A)1 - P(X lt A)
• This is useful because the table only gives
  probabilities for events such as (XltA) not
  (XgtA).

11
How do we actually solve problems

• Draw.
• Indicate the area you wish to find.
• Convert the probability you are interested in to
  z-scores.
• This is because every normally distributed random
  variable transforms to the standard normal so
  only the standard normal is in your book.
• Manipulate (use the tricks)
• P(altzltb)P(zltb)-P(zlta)
• P(zgta)P(zlt-a)

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Why is the Normal Distribution so important

• Almost every distribution you examine will be
  closely related to the normal distribution.
• Consider the Empirical Rule.
• Almost every random variable can be transformed
  into another that is approximately normal.
• It is (relatively) easy to calculate
  probabilities for normally distributed random
  variables by using the Standard Normal
  distribution.

13
Conclusion

• Objectives addressed
• Calculate probabilities based on the Normal
  Distribution.
• Explain why the Normal Distribution is important.
• Problems
• 4.61a 4.62 a d 4.65 a e 4.69 4.72
• Exam 1A 27 through 33