biostatist

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 tendencyPopulation standard deviation (σ) as a measure of dispersion

The entire family of normal probability distributions is defined by its mean m and its standard deviation s .
Normal Probability Distribution
Characteristics
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

The distribution is symmetric, and is bell-shaped.

Normal Probability Distribution
Characteristics
x

The highest point on the normal curve is at the mean, which is also the median and mode.

Normal Probability Distribution
Characteristics
x


The Normal Distribution “Gaussian Distribution” The mean, median, and the mode are equal
Mean=Median=Mode
Total P=1

Normal Probability Distribution

Characteristics
-10
0
20
The mean can be any numerical value: negative, zero, or positive.
x



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

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).
Normal Probability Distribution
Characteristics
.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 axisThe 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)

8.* Normal Distribution…

Normal Probability Distribution
Characteristics
s = 15
s = 25
The standard deviation determines the width of the curve: larger values result in wider, flatter curves.
x



The Normal Distribution “Gaussian Distribution” Different values of (σ) determine the degree of flatness or peakedness of the graph of the distributionWhen (σ) is increased the curve will be more flatWhen (σ) is decreased the curve will be more peaked

8.* Normal Distribution…

The Normal Distribution “Gaussian Distribution” µ ± 1 σ 68% of the areaµ ± 2 σ 95% of the areaµ ± 3 σ 99.7% of the area

The Normal Distribution “Gaussian Distribution” µ ± 1 σ 68% of the area

The Normal Distribution “Gaussian Distribution” µ ± 2 σ 95% of the area

The Normal Distribution “Gaussian Distribution” µ ± 3 σ 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


s = 1
0
z
The letter z is used to designate the standard normal random variable.
Standard Normal Probability Distribution

Exercise

Find for a standard normal distribution P(0< Z <1.2) P(Z >1.2) P(-1.2< Z <1.2) P(Z <-1.2 or Z >1.2) P(Z <1.2) 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 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)

80 mmHg

10 mmHg
X
90
100
110
50
60
70

Exercise

If the weight of 6-years old boys is normallydistributed with µ =25 Kg, and σ = 2 kg. Find:P(20< X <25)P(X >28)P(X >22)P(X <22)P(X <28)P(26< X <29)

25Kg

2 Kg
X
27
29
31
19
21
23