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PROBABILITY
DISTRIBUTION
PROBABILITY DISTRIBUTION
Probability Distribution of
a Continuous Variable
The Normal Distribution
“Gaussian Distribution”



Is a theoretical model that has been found
to fit many naturally occurring
phenomena.
It is the most important distribution in
statistics
It is used for continuous variables
The Normal Distribution
“Gaussian Distribution”



The parameters in this distribution are the:
Population mean (µ) as a measure of central
tendency
Population standard deviation (σ) as a measure
of dispersion
Normal Probability Distribution
Characteristics
The entire family of normal probability
distributions is defined by its mean m and its
standard deviation s .
Standard Deviation s
Mean m
x
The Normal Distribution
“Gaussian Distribution”


The curve is symmetric around the
mean
The total area under the curve equal
one
Normal Probability Distribution
Characteristics
The distribution is symmetric, and is bell-shaped.
x
Normal Probability Distribution
Characteristics
The highest point on the normal curve is at the
mean, which is also the median and mode.
x
The Normal Distribution
“Gaussian Distribution”

The mean, median, and the mode
are equal
Mean=Median=Mode
Total P=1
Normal Probability Distribution
Characteristics
The mean can be any numerical value: negative,
zero, or positive.
x
-10
0
20
The Normal Distribution
“Gaussian Distribution”

50% of the area under the curve is
on the right side of the curve and the
other 50% is on its left
Normal Probability Distribution
Characteristics
Probabilities for the normal random variable are
given by areas under the curve. The total area
under the curve is 1 (.5 to the left of the mean and
.5 to the right).
.5
.5
x
The Normal Distribution
“Gaussian Distribution”
With fixed (σ) different values of µ will shift the
graph of the distribution along the X axis
 The shape of the curve will not changed, but it
will be shifted to:
the right ( when µ is increased)
or to the left (when µ is decreased)

Normal Distribution…
8.15
Normal Probability Distribution
Characteristics
The standard deviation determines the width of the
curve: larger values result in wider, flatter curves.
s = 15
s = 25
x
The Normal Distribution
“Gaussian Distribution”



Different values of (σ) determine the degree of
flatness or peakedness of the graph of the distribution
When (σ) is increased the curve will be more flat
When (σ) is decreased the curve will be more peaked
Normal Distribution…
8.18
Normal Densities
0.045
0.04
0.035
0.03
N(100,400)
0.025
f(y)
N(100,100)
N(100,900)
N(75,400)
0.02
N(125,400)
0.015
0.01
0.005
0
0
20
40
60
80
100
y
120
140
160
180
200
The Normal Distribution
“Gaussian Distribution”

µ±1σ
68% of the area

µ±2σ
95% of the area

µ±3σ
area
99.7% of the
The Normal Distribution
“Gaussian Distribution”

µ±1σ
68% of the area
The Normal Distribution
“Gaussian Distribution”

µ±2σ
95% of the area
The Normal Distribution
“Gaussian Distribution”

µ±3σ
99.7% of the area
68-95-99.7 Rule
68% of
the data
95% of the data
99.7% of the data
The unit normal , or the Standard
normal distribution
X- µ
Z= --------σ
Standard Normal Probability Distribution
The letter z is used to designate the standard
normal random variable.
s=1
z
0
Exercise
Find for a standard normal distribution
a) P(0< Z <1.2)
b) P(Z >1.2)
c)
P(-1.2< Z <1.2)
d) P(Z <-1.2 or Z >1.2)
e) P(Z <1.2)
f)
P(1.5 < Z <2.0)
Exercise
If
µ of DBP of a population = 80 mmHg, and
σ2 =100(mmHg)2 .What is the probability
1)
2)
3)
4)
5)
6)
of selecting a man with DBP of:
P(75< X < 85)
P(60< X <100)
P(65< X <95)
P(X <60)
P(X >100)
P(90< X <100)
10 mmHg
50 60
70
90 100 110
80 mmHg
X
Exercise
If the weight of 6-years old boys is normally
distributed with µ =25 Kg, and
Find:
1.
P(20< X <25)
2.
P(X >28)
3.
P(X >22)
4.
P(X <22)
5.
P(X <28)
6.
P(26< X <29)
σ = 2 kg.
2 Kg
19 21 23
27
29 31
25Kg
X