ECON 4630
ECON 5630
TOPIC #5: CONTINUOUS PROBABILITY DISTRIBUTIONS
I.
Continuous Probability Distributions in General
A.
Probability Density
Relative
frequency
density
Recall our example involving height:
Class
midpoint
f
f/n
152
2
0.01
159
4
0.02
166
12
0.06
173
44
0.22
180
64
0.32
187
56
0.28
194
16
0.08
201
2
0.01
f
n
class width
1.00
1
f
f/n
60
0.30
40
0.20
20
0.10
152
159
166
173
180
187
194
201
rel. freq. density
If I now graph with relative frequency density on the vertical axis, I get a rescaled version of this picture:
0.04
0.025
0.01
152
159
166
173
180
187
194
201
2
What is the total area of the shaded region?
B.
Continuous Probability Densities
What happens when we increase the sample size and decrease the class interval?
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II.
III.
Mean and Variances of Continuous-Distributed Variables
A.
Mean of a Continuous-Distributed Variable
B.
Variance of a Continuous-Distributed Variable
The Standard Normal Distribution
A.
General information
Many variables have as their probability distribution the normal or Gaussian curve.
This is the familiar bell curve.
B.
The Standard Normal
1.
Definition: A random variable Z is distributed standard normal if its
probability density function is:
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2.
A crude plot:
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3.
Characteristics of the Standard Normal
4.
Mean and Variance of Standard Normal
5.
Using the standard normal table: examples
Example #1: What is the probability that Z exceeds 1.6?
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Example #2: What is the probability that Z is between –1 and +1?
Example #3: P(Z < -1.81)
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Example #4: P(-1.11 < Z < 0.19)
Example #5: P(0.35 < Z < 1.17)
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IV.
The General Normal
A.
Distinction between standard and general normal
The standard normal is a very specialized distribution. It has a mean of 0 and a
standard deviation of 1. The general normal can have any mean and standard
deviation
B.
Definition: A random variable X is distributed general normal with a mean of μ
and a standard deviation of σ if its probability density function is:
C.
Using the standard normal table with a general normal variable:
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D.
Examples
Example #1: A bolt picked at random from a production line has length X. X is
distributed normally with  = 78.3 millimeters and  = 1.4 millimeters. If all
bolts longer than 80 millimeters have to be discarded, what proportion of output is
wasted?
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Example #2: The shape of a frequency curve showing the length of drive, D, for a
particular golf pro is close enough to be normal to be treated as such. The mean
drive is 237 yards with a standard deviation of 45 yards. If the pro hits 8,000
drives in the course of a year, how many of them would you predict will be
between 256 and 290 yards?
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Example #3: Prof. Nasty-Domy has, shall we say, a limited social life (this is an
occupational hazard among economists and others, including dentists and
morticians). To fill his lonely and pathetic evenings, he plays the handheld
Yahtzee game that his granny gave him back in 1996. He has discovered that over
the many thousands of games he has played, his average score is 237, with a
variance of 2000.
(a)
What is the probability that he will score more than 400?
(b)
What is the probability that he will score less than 200?
(c)
What is the probability that his score will be somewhere between
250 and 300?
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Example #4: Working “Backwards:”
So far, we’ve figured out the probability based on some X or Z value. However,
sometimes we need to know the X or Z value based on the probability.
Example #3: Suppose the average height of men is 180 cm, with a standard
deviation of 8. What is the 99th height percentile? (OR, 99% of men are at least
what height?)
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Example #5: The average birthweight for babies at a certain hospital is 6.78
pounds, with a standard deviation of 0.45 pounds. Treat as normal the frequency
curve that shows the distribution of these weights.
a)
What is the probability that a baby born at this hospital has a
weight between 6 and 7 pounds?
b)
What percentage of babies born weigh at least 8 pounds?
c)
Unusually small babies are given special attention. If the hospital
defines “unusually small” as having a birth weight in the first
quartile, what is the maximum birth weight that merits special
attention?
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E.
Excel Commands
=NORMDIST(x, μx, σx, cumulative)
Where μx is the mean
σx is the standard deviation
cumulative is “TRUE” for the cumulative distribution
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NOTES:
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