Continuous Probability Distributions (The Normal Distribution-I)

1
Continuous Probability Distributions(The Normal
Distribution-I)

• QSCI 381 Lecture 16
• (Larson and Farber, Sect 5.1-5.3)

2
Introduction

• The (or Gaussian) distribution is
  probably the most used (and abused) distribution
  in statistics.
• Normal random variables are continuous (they can
  take any value on the real line) so the Normal
  distribution is an example of a continuous
  probability distribution.

Normal
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The Normal Distribution-I
Inflection points
Total area1
?
?-?
?-2?
?-3?
?3?
?2?
??
The graph of the normal distribution is called
the normal (or bell) curve.
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The Normal Distribution-II

• The mean, median, and mode are the same.
• The normal curve is symmetric about its mean.
• The total area under the normal curve is one.
• The normal curve approaches, but never touches,
  the x-axis.

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The Normal Distribution-III

• Notation
• We say that the random variable X is normally
  distributed with mean ? and standard deviation ?
• The equation defining the normal curve is
• f (x) is referred to as a

probability
(pdf).
density function
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Normal Distributions and Probability-I

• We do not talk about the probability PXx for
  continuous random variables. Rather we talk about
  the probability that the random variable falls in
  an interval, i.e. Px1 ? X ? x2.
• Px1 ? X ? x2 can be determined by finding the
  area under the normal curve between x1 and x2.

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Normal Distributions and Probability-II
Of the total area under the curve about 68
lies between ?-? and ?? 95 lies between ?-2?
and ?2? 99.7 lies between ?-3? and ?3?
68
34
95
13.5
99.7
µ
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Example

• The mean length of a fish of age 5 is 40cm and
  the standard deviation is 4. What is the
  probability that the length of a randomly
  selected fish of age 5 is less than 48cm?
• 48cm is two standard
• deviations above the
• mean so the area
• to the left of 48cm is
• 0.50.475 0.975.

This is an approximate method we will develop a
more accurate method later.
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The Standard Normal Distribution-I

• The normal distribution with a mean of 0 and a
  standard deviation of 1 is called the
• The standard normal distribution is closely
  related to the z-score

standard normal distribution
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The Standard Normal Distribution-II
Area1
The area under the standard normal
distribution from -? to x is often listed in
tables. can be computed using the EXCEL
function NORMDIST.
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Example-I

• Find the area under the standard normal
  distribution between -0.12 and 1.23.
• We use our previous approach
• P-0.12 ? X ? 1.23 PX?1.23 - PX?-0.12
• In EXCEL
• NORMDIST(1.23,0,1,TRUE)-NORMDIST(-0.12,0,1,TRUE)
• NORMDIST(x,?,?,cum)

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Example-II
PX?1.23
PX?-0.12
PX?1.23-PX?-0.12
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Further Examples

• How would you find the probability that
• X ? -0.95
• X ? 1
• 0.1 ? X ? 0.5 OR 1.5 ? X ? 2
• 0.1 ? X ? 0.5 AND 1.5 ? X ? 2

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Generalizing-I

• We can transform any normal distribution into a
  standard normal distribution by subtracting the
  mean and dividing by the standard deviation.

?300 ?20x330
?0 ?1z1.5
Area0.933 in both cases
Z(x-300)/20
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Generalizing-II

• To find the probability that X ?Y if X is
  normally distributed with mean ? and standard
  deviation ?.
• Compute the z-score z (Y - ?)/?
• Calculate the area under the normal curve between
  -? and z using tables for the standard normal
  distribution.
• We could calculate this area directly using the
  EXCEL function NORMDIST.

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Examples-1

• The average swimming speed of a fish population
  is 2m.s-1 (standard deviation 0.5). You select a
  fish at random. What is the probability that
• Its swimming speed is less than 1m.s-1.
• Its swimming speed is greater than 2.5m.s-1.
• Its swimming speed is between 2 and 3 m.s-1.

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Examples-1

• The average swimming speed of a fish population
  is 2m.s-1 (standard deviation 0.5). You select a
  fish at random. What is the probability that
• Its swimming speed is less than 1m.s-1.
• Its swimming speed is greater than 2.5m.s-1.
• Its swimming speed is between 2 and 3 m.s-1.
• P(z lt (1-2)/.5) P(z lt -2) 0.0228

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Examples-1

• The average swimming speed of a fish population
  is 2m.s-1 (standard deviation 0.5). You select a
  fish at random. What is the probability that
• Its swimming speed is less than 1m.s-1.
• Is swimming speed is greater than 2.5m.s-1.
• Its swimming speed is between 2 and 3 m.s-1.
• P(z gt (2.5-2)/.5) P(z gt 1) 1 - P(z 1)
  0.159

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Examples-1

• The average swimming speed of a fish population
  is 2m.s-1 (standard deviation 0.5). You select a
  fish at random. What is the probability that
• Its swimming speed is less than 1m.s-1.
• Its swimming speed is greater than 2.5m.s-1.
• Its swimming speed is between 2 and 3 m.s-1.
• P( (2-2)/.5 z (3-2)/0.5 ) P(0 z 2)
  P(z 2) P(z 0) 0.477

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Examples-1

• The average swimming speed of a fish population
  is 2m.s-1 (standard deviation 0.5). You select a
  fish at random. What is the probability that
• Its swimming speed is less than 1m.s-1.
• Its swimming speed is greater than 2.5m.s-1.
• Its swimming speed is between 2 and 3 m.s-1.
• Why is the normal distribution not entirely
  satisfactory for this case?

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Examples-2

• In a large group of men 4 are under 160cm tall
  and 52 are between 160 cm and 175 cm. Assuming
  that the heights of men are normally distributed,
  what are the mean and standard deviation of the
  distribution?

22
Examples-2

• In a large group of men 4 are under 160cm tall
  and 52 are between 160 cm and 175 cm. Assuming
  that the heights of men are normally distributed,
  what are the mean and standard deviation of the
  distribution?
• P(z lt (160-µ)/s ) 0.04 P( z lt -1.75)
• P( (160-µ)/s z (175-µ)/s ) 0.52
• 0.56 P(z (175-µ)/s ) P(z 0.15 )
• So -1.75 (160-µ)/ s 0.15 (175- µ)/ s
  µ 174 , s 8

We get these from a standard normal table
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Examples-3

• An anchovy net can hold 10 t of anchovy. The mean
  mass of anchovy school is 8 t with a standard
  deviation of 2 t. What proportion of schools
  cannot be fully caught?
• Hint we are interested in the proportion of
  schools greater than 10t.
• ie. 1- P(X 10) 1- P(z (10-8)/2)
• OR 1 - NORMDIST(10,8,2,TRUE)
  0.159