Econ 3790: Business and
Economics Statistics
Instructor: Yogesh Uppal
Email: yuppal@ysu.edu
Chapter 6
Continuous Probability Distributions

Normal Probability Distribution
p(x)
Normal
x
Normal Probability Distribution


The normal probability distribution is the
most important distribution for describing a
continuous random variable.
It is widely used in statistical inference.
Normal Probability Distribution

It has been used in a wide variety of applications:
Heights
Heights
of
ofpeople
people
Scientific
Scientific
measurements
measurements
Normal Probability Distribution

It has been used in a wide variety of applications:
Test
Test
scores
scores
Amounts
Amounts
of
of rainfall
rainfall
Normal Distributions

The probability of the random variable assuming a
value within some given interval from x1 to x2 is
defined to be the area under the curve between x1
and x2.
f (x)
Normal
x1 x2
x
Normal Probability Distribution

Characteristics
The
The distribution
distribution is
is symmetric
symmetric;; its
its skewness
skewness
measure
measure is
is zero.
zero.
x
Normal Probability Distribution

Characteristics
The
The highest
highest point
point on
on the
the normal
normal curve
curve is
is at
at the
the
mean
mean,, which
which is
is also
also the
the median
median and
and mode
mode..
Mean = 
x
Normal Probability Distribution

Characteristics
The
The entire
entire family
family of
of normal
normal probability
probability
distributions
distributions is
is defined
defined by
by its
its mean
mean  and
and its
its
standard
standard deviation
deviation  ..
Standard Deviation 
Mean 
x
Normal Probability Distribution

Characteristics
The
The mean
mean can
can be
be any
any numerical
numerical value:
value: negative,
negative,
zero,
zero, or
or positive.
positive. The
The following
following shows
shows different
different normal
normal
distributions
distributions with
with different
different means.
means.
-10
0
20
x
Normal Probability Distribution

Characteristics
The
The standard
standard deviation
deviation determines
determines the
the width
width of
of the
the
curve:
curve: larger
larger values
values result
result in
in wider,
wider, flatter
flatter curves.
curves.
 = 15
 = 25
x
Same Mean
Normal Probability Distribution

Characteristics
Probabilities
Probabilities for
for the
the normal
normal random
random variable
variable are
are
given
given by
by areas
areas under
under the
the curve
curve.. The
The total
total area
area
under
under the
the curve
curve is
is 11 (.5
(.5 to
to the
the left
left of
of the
the mean
mean and
and
.5
.5 to
to the
the right).
right).
.5
.5
Mean 
x
Standardizing the Normal Values or the
z-scores

Z-scores can be calculated as follows:
z
x

•We can think of z as a measure of the number of
standard deviations x is from .
Standard Normal Probability Distribution
A
n of
A standard
standard normal
normal distribution
distribution isis aa normal
normal distribution
distribution with
with mea
mean
of
00 and
μ) and
and variance
variance of
of 1.
1. IfIf xx has
has aa normal
normal distribution
distribution with
with mean
mean ((μ)
and
Variance
σ), then
Variance ((σ),
then zz isis said
said to
to have
have aa standard
standard normal
normal distribution
distribution..

0
z
Example: Air Quality



I collected this data on the air quality of various
cities as measured by particulate matter index
(PMI). A PMI of less than 50 is said to
represent good air quality.
The data is available on the class website.
Suppose the distribution of PMI is
approximately normal.
Example: Air Quality


The mean PMI is 41 and the standard
deviation is 20.5.
Suppose I want to find out the probability that
air quality is good or what is the probability
that PMI is greater than 50.