1
Chapter 6: Probability
Distributions
Read Chapter 7
Random Variables Definitions
A
Random Variable
,
X
, is a real valued function whose
values are determined by a random experiment.
Mean:
E(X)=
μ
=
Σ
x p(x)
all x
Variance:
E[(X
μ
)
2
]=
σ
2
=
Σ
(x
μ
)
2
p(x)
all x
Mass function:
P(X=x) = p(x)
Standard Deviation =
σ
The Normal Distribution
The
normal distribution
is often called the Gaussian
distribution,
after Carl Friedrich Gauss, who discovered many of
its properties.
Gauss, commonly viewed as one of the greatest
mathematicians of all time (if not the greatest), was honored by
Germany on their 10 Deutschmark bill. The Gaussian
distribution is the probabilistic model for much of this course.
2
2
2
)
(
2
2
2
1
)
,

(
σ
μ
πσ
σ
μ
−
−
=
x
e
x
f
Normal Distribution
Normal distributions are
“Bell shaped”
Symmetric around the mean
The mean (
μ
) and the standard deviation (
σ
)
completely describe the density curve
Increasing/decreasing
μ
moves the curve along the
horizontal axis
Increasing/decreasing
σ
controls the spread of the
curve
xx
Density
100
200
300
400
500
0.0
0.02
0.04
0.06
0.08
Different Means and Variances
ZScores and the Standard Normal Distribution
The zscore for a value
x
of a random variable is
the number of standard deviations that
x
falls
from the mean
A negative (positive) zscore indicates that the
value is below (above) the mean
zscores can be used to calculate the
probabilities of a normal random variable using
the normal tables in the back of the book
z
=
x
−
μ
σ
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
ZScores and the Standard Normal Distribution
A standard normal distribution
has mean
μ=0
and standard deviation
σ
=1
When a random variable has a normal
distribution and its values are converted
to
z
scores by subtracting the mean and
dividing by the standard deviation, the
z

scores have the standard normal
distribution.
Table A: Standard Normal Probabilities
Table A enables us to find normal probabilities
It tabulates the normal cumulative probabilities falling
below
the point
μ
+z
σ
To use the table:
Find the corresponding
z
score
Look up the closest standardized score (
z
) in the table.
First column gives
z
to the first decimal place
First row gives the second decimal place of
z
The corresponding probability found in the body of the
table gives the probability of falling
below
the
z
score
Using Table A
Find the probability that a normal random
variable takes a value less than 1.43 standard
deviations above
μ; P(Z<1.43)=.9236
Stat Crunch to the Rescue!!
Using Table A
Find the probability that a normal random variable
takes a value greater than 1.43 standard deviations
above
μ: P(Z>1.43)=1.9236=.0764
3
Find the probability that a normal random
variable assumes a value within 1.43 standard
deviations of
μ
Probability below
1.43
σ
=
.9236
Probability below
1.43
σ
= .0764 (=1.9236)
P(1.43<Z<1.43) =.9236.0764=.8472
P(1.43<Z<1.43) = .9236.0764 = .8472
How Can We Find the Value of
z
for a Certain
Cumulative Probability?
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 '07
 VELLEMANP
 Normal Distribution, Probability theory, systolic blood pressure

Click to edit the document details