The Normal Distribution

1
Section 4.3

• The Normal Distribution

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The standard normal random variable.

• A continuous random variable whose distribution
  is given by the following pdf/cdf is called a
  standard normal random variable.
• It is clear from the graph of the pdf that the
  mean of a standard normal random variable is 0.
  One can also show that the variance is 1.
• A standard normal random variable is called
  N(0,1) and is denoted by Z (instead of X).

3
The probability density function.

• The function that describes the pdf1 of a
  standard normal random variable is
• The maximum of this function (where the first
  derivative is 0) occurs at x0.
• The inflection points (where the second
  derivative is 0) occur at 1.
• Practically all of the area under the curve is
  between 3.

1Of course, one should verify that this function
is actually a pdf, that is the area under the
curve is one. However, this function is one of
those rare functions that does not have an
anti-derivative. One has to use numerical
integration or a trick involving complex
variables to show that the area is one and so we
will omit this.
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Finding Probabilities

• Let Z be the standard normal random variable.
  In order to evaluate P(a ? Z ? b) , evaluate
  F(b)- F(a), where F(z) is the cdf of Z and is
  stored in table A.3.
• Calculate P(Z ? 1).
• Calculate P(Z?1).
• Calculate P(-0.5 ? Z ? 2).
• Note, you can not use the pdf directly, because
  the pdf has no antiderivative.

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Percentiles and Critical Values of Standard
Normal Distribution

• Use Table A.3 to find the 95th percentile of the
  standard normal random variable.
• Note that area to the left of thevalue you just
  found is 0.95, and area to the right is 0.05.
  Thus ____ is also called the z0.05 critical
  value.
• In general, the za critical value is the number
  such that P(Zgt za) a. Common critical values
  are given in Table 4.1, pg 163.

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One more example on Z.

• Find c so that P(Z gtc)0.05.
• Find c so that P(Z gt c)0.05.
• You do not have to interpolate. Give the z score
  that is closer. If desired probability is
  exactly halfway in between two z sccores , give
  the average z score.

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Normal Random Variables.

• Let ?, ? be any real numbers with ? gt 0. A
  continuous random variable X with outcome space
  (??,?) and pdf
  1 f (x) ????
  exp ?( x?? )2/ (2?2)
  ? ? 2? is said to have a normal
  distribution with mean ? and standard deviation
  ?. Sometimes we say that X has an N(?,?2)
  distribution.
• Picture
• Recall empirical rule that was presented in last
  weeks lab.

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Some Normal PDFs
N(0,1/4)
N(3,1)
N(0,1)
N(0,4)
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Standardizing

• Proposition Let X be an N(?,?2) random variable.
  Then the random variable
• Z ( X ? ? )/? is N(0,1).
• Suppose X is N(10,9). Calculate
• P(X ? 10)
• P(0 ? X ? 15)
• Find the 95th percentile ofX.
• Find c so that P(X-10gtc.05)

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An example

• The amount of distilled water dispensed by a
  machine is normally distributed with mean 64 oz
  and standard deviation 0.78 oz.
• What is the probability that a container holding
  65 oz will overflow?
• How small should the standard deviation be so
  that only 1 of containers holding 65 oz overflow?

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Normal approximation to Binomial

• If np and n(1-p) are both greater than or equal
  to 10, the the binomial pmf is well approximated
  by the normal pdf (using a continuity
  correction).
• Example. When circuit boards used in the
  manufacture of cd players are tested, the long
  run percentage of defectives is 5. Suppose that
  a batch of 250 boards has beed received and that
  the condition of any particular board is
  indepenndent of the other 249.
• What is the approximate probability that at least
  10 of the boards in the batch are defective?
• What is the approximate probability that there
  are exactly 10 defective boards?
• More in lab tomorrow.