Properties of the Normal Distribution

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Lesson 7 - 1

• Properties of the Normal Distribution

2
Quiz

• Homework Problem Chapter 6 reviewThe number of
  cars that arrive at a banks window between 300
  pm and 6 pm on Friday follows a Poisson process
  at the rate of 0.41 car every minute. Compute
  the possibility that the number of cars that
  arrive at the bank between 400 pm and 410 pm
  isa) exactly four carsb) fewer than four
  carsc) at least four cars
• Reading questions
• What does the area under the graph represent in a
  continuous PDF?
• What is the standard normal distribution variable
  called?

3
Objectives

• Understand the uniform probability distribution
• Graph a normal curve
• State the properties of the normal curve
• Understand the role of area in the normal density
  function
• Understand the relationship between a normal
  random variable and a standard normal random
  variable

4
Vocabulary

• Continuous random variable has infinitely many
  values
• Uniform probability distribution probability
  distribution where the probability of occurrence
  is equally likely for any equal length intervals
  of the random variable X
• Normal curve bell shaped curve
• Normal distributed random variable has a PDF or
  relative frequency histogram shaped like a normal
  curve
• Standard normal normal PDF with mean of 0 and
  standard deviation of 1 (a z statistic!!)

5
Uniform PDF

• Sometimes we want to model a random variable that
  is equally likely between two limits
• When every number is equally likely in an
  interval, this is a uniform probability
  distribution
• Any specific number has a zero probability of
  occurring
• The mathematically correct way to phrase this is
  that any two intervals of equal length have the
  same probability
• Examples
• Choose a random time the number of seconds past
  the minute is random number in the interval from
  0 to 60
• Observe a tire rolling at a high rate of speed
  choose a random time the angle of the tire
  valve to the vertical is a random number in the
  interval from 0 to 360

6
Discrete Uniform PDF
P(x0) 0.25 P(x1) 0.25 P(x2) 0.25 P(x3)
0.25
Continuous Uniform PDF
P(x1) 0 P(x 1) 0.33 P(x 2) 0.66 P(x
3) 1.00
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Probability in a Continuous Probability
Distributions

• Let P(x) denote the probability that the random
  variable X equals x, then
• 1) ? P(x) 1 (sum of all probabilities must
  equal 1) ? total area under the PDF graph must
  equal 1
• 2) The probability of x occurring in any
  interval, P(x), must between 0 and 1 0
  P(x) 1 ? the height of the graph of the PDF
  must be greater than or equal to 0 for all
  possible values of the random variable
• 3) The area underneath probability density
  function over some interval represents the
  probability of observing a value of the random
  variable in that interval.

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Properties of the Normal Density Curve

• It is symmetric about its mean, µ
• Because mean median mode, the highest point
  occurs at x µ
• It has inflection points at µ s and µ s
• Area under the curve 1
• Area under the curve to the right of µ equals the
  area under the curve to the left of µ, which
  equals ½
• As x increases or decreases without bound (gets
  farther away from µ), the graph approaches, but
  never reaches the horizontal axis (like
  approaching an asymptote)
• The Empirical Rule applies

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Normal Curves

• Two normal curves with different means (but the
  same standard deviation) on left
• The curves are shifted left and right
• Two normal curves with different standard
  deviations (but the same mean) on right
• The curves are shifted up and down

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Empirical Rule
µ 3s
µ 2s
µ s
99.7
95
68
2.35
2.35
34
34
13.5
13.5
0.15
0.15
µ
µ - 2s
µ 2s
µ - s
µ - 3s
µ s
µ 3s
Normal Probability Density Function
1 y -------- e v2p
-(x µ)2 2s2
where µ is the mean and s is the standard
deviation of the random variable x
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Area under a Normal Curve

• The area under the normal curve for any interval
  of values of the random variable X represents
  either
• The proportion of the population with the
  characteristic described by the interval of
  values or
• The probability that a randomly selected
  individual from the population will have the
  characteristic described by the interval of
  values the area under the curve is either a
  proportion or the probability

12
Standardizing a Normal Random Variable

• our Z statistic from before
• X - µ
• Z -----------
  
• s
• where µ is the mean and s is the standard
  deviation of the random variable X
• Z is normally distributed with mean of 0 and
  standard deviation of 1
• Note we are going to use tables (for Z
  statistics) not the normal PDF!!
• Or our calculator (see next chart)

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Normal Distributions on TI-83

• normalpdf     pdf Probability Density
  FunctionThis function returns the probability of
  a single value of the random variable x.  Use
  this to graph a normal curve. Using this function
  returns the y-coordinates of the normal curve.
• Syntax   normalpdf (x, mean, standard
  deviation)taken from http//mathbits.com/MathBit
  s/TISection/Statistics2/normaldistribution.htm

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Normal Distributions on TI-83

• normalcdf    cdf Cumulative Distribution
  FunctionThis function returns the cumulative
  probability from zero up to some input value of
  the random variable x. Technically, it returns
  the percentage of area under a continuous
  distribution curve from negative infinity to the
  x.  You can, however, set the lower bound.
• Syntax  normalcdf (lower bound, upper bound,
  mean, standard deviation)(note lower bound is
  optional and we can use -E99 for negative
  infinity and E99 for positive infinity)

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Normal Distributions on TI-83

• invNorm     inv Inverse Normal PDFThis
  function returns the x-value given the
  probability region to the left of the x-value.
  (0 lt area lt 1 must be true.)  The inverse normal
  probability distribution function will find the
  precise value at a given percent based upon the
  mean and standard deviation.
• Syntax  invNorm (probability, mean, standard
  deviation)

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Example 1

• A random number generator on calculators randomly
  generates a number between 0 and 1. The random
  variable X, the number generated, follows a
  uniform distribution
• Draw a graph of this distribution
• What is the P(0ltXlt0.2)?
• What is the P(0.25ltXlt0.6)?
• What is the probability of getting a number gt
  0.95?
• Use calculator to generate 200 random numbers

0.20
0.35
0.05
Math ? prb ? rand(200) STO L3 then 1varStat L3
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Example 2

• A random variable x is normally distributed with
  µ10 and s3.
• Compute Z for x1 8 and x2 12
• If the area under the curve between x1 and x2 is
  0.495, what is the area between z1 and z2?

8 10 -2 Z ---------- -----
-0.67 3 3
12 10 2 Z ----------- -----
0.67 3 3
0.495
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Summary and Homework

• Summary
• Normal probability distributions can be used to
  model data that have bell shaped distributions
• Normal probability distributions are specified by
  their means and standard deviations
• Areas under the curve of general normal
  probability distributions can be related to areas
  under the curve of the standard normal
  probability distribution
• Homework
• pg 367 371 7 12 15-16, 19-20, 32-33