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Principles of Biostatistics
2nd Edition
2nd
Edition
by Marcello Pagano & Kimberlee Gauvreau
2000
Chapter 7
Theoretical Probability
Theoretical Probability Distributions
Liang‐‐Yi Wang Liang
Yi Wang
Department of Public Health, NCKU
[email protected]
2010/3/17
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Contents
Probability distributions
Probability
distributions
 The binominal distribution
 The Poisson distribution
 The normal distribution
The normal distribution

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V i bl
Variable
A characteristic that can be measured or categorized
Random variable
A variable can assume a number of different value such that any particular outcome is determined by chance
any particular outcome is determined by chance

Discrete random variable
◦ Only a finite or countable number of outcomes. Ex: marital status( single, married, divorced, widowed), the number of infections an infant develops during his or her first year of
infections an infant develops during his or her first year of life. 
Continuous random variable
◦ A variable can take on any value within a specified interval continuum, ex: weight, height.
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Probability Distribution
Probability Distribution

A probability distribution applies the A
probability distribution applies the
theory of probability to describe the behavior of the random variable.
f
◦ Discrete random variable
 Determine the probabilities of possible outcomes of the random variable
◦ Continuous random variable
 determine the probabilities associated with p
specified ranges of values.
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Discrete random variable
• Determine the probabilities of p
possible outcomes of the random variable
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Continuous random variable
• determine the probabilities p
associated with specified ranges of values.
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Probability Distribution
Probability Distribution

Empirical probability distribution
Empirical probability distribution
◦ Probability that are calculated from a finite amount of data

Theoretical probability distribution
p
y
◦ Probability that are determined based on theoretical considerations.
theoretical considerations.
◦ N, mean, variance, …..
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The binominal distribution
Th
bi
i l di t ib ti
The Poisson distribution
The normal distribution
THEORETICAL PROBABILITY DISTRIBUTION
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Theoretical Probability Distribution
What have to Know?
Function of P(X), mean, variance, sd
Function
of P(X) mean variance sd
2. Parameter 3. The probability distribution
1
1.
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THE BINOMINAL DISTRIBUTION
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What is the Binominal Distribution?
What is the Binominal Distribution?

Dichotomous random variable (Y)
Dichotomous random variable (Y)
◦ Live and death; male and female; sickness and health; failure and success
◦ Bernoulli random variable
29% of the adults in the US are smoker (Y=1, smoker; Y=0, nonsmoker)
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What is the Binominal Distribution
What is the Binominal Distribution

Dichotomous random variable (Y)
Dichotomous random variable (Y)
◦ Live and death; male and female; sickness and health; failure and success
◦ Bernoulli random variable ( two mutually exclusive outcomes)

The number of persons in the sample who are smokers
(X) ◦ Ex: sample size n=2, X can take on 0, 1, or 2. ◦ Bernoulli trial
B
lli t i l
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n=2
What is the Binominal Distribution
What is the Binominal Distribution
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n=2
What is the Binominal Distribution
What is the Binominal Distribution
probability
0.504
0.412
0.084
0
0.6
6
Probability X=x
0.5
Binomial distribution (n=2 p=0 29)
(n=2, p=0.29)
0.4
0.3
0.2
0.1
0
0
1
2
Number of smokers x
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Assumption (the Binominal Distribution)
(the Binominal Distribution)
There are a fixed number of trials n, each of which results in one of two
each of which results in one of two mutually exclusive outcomes.
2) The outcomes of the n
Th
f h
trials are i l
independent.
3) The probability of success p is constant for each trial
for each trial.
1)
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n=3 p=0 29
n=3, p=0.29
The Binominal Distribution
The Binominal Distribution
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Parameter = n, p
The Binominal Distribution
The Binominal Distribution
Function of P(X),
mean, variance, SD
i
Mean= np
Variance= np(1‐p)
Variance= np(1‐p)
SD= Try to use Try to use Table A.1 Table A.1 in Appendix A
in Appendix A
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n=10 p=0 29
n=10, p=0.29
The Binominal Distribution
The Binominal Distribution
Mean= np
Mean
np=2.9
2.9
SD= 2010/3/17
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SD= 2010/3/17
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(review)
Th
Theoretical Probability Distribution
ti l P b bilit Di t ib ti
What have to Know?
Function of P(X), mean, variance, sd
2. Parameter 3 The probability distribution
3.
The probability distribution
1.
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THE POISSON DISTRIBUTION
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What is the Poisson Distribution
What is the Poisson Distribution

Ex:
Ex: ◦ We are interested in the probability distribution of the number of individuals involved in a motor vehicle
the number of individuals involved in a motor vehicle accident each year (x) in a town of the US.
◦ p=0.00024, n=10,000 Difficult to calculate using binomial dist.!!!
using binomial dist !!!
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What is the Poisson Distribution
What is the Poisson Distribution


When n is very large and p is very small, the When
n is very large and p is very small the
binomial distribution is well approximated by another theoretical probability distribution
another theoretical probability distribution
‐‐‐‐‐‐‐Poisson distribution
Poisson distribution
Th P i
The Poisson distribution is used to model di t ib ti i
dt
d l
discrete events that occur infrequently in time or space.
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The Poisson Distribution
The Poisson Distribution
A
Assumption
ti
1)
2)
3)
The probability that a single event occurs within an interval is proportional to the length h
l
l
h l
h
of the interval
Wi hi
Within a single interval, an infinite number of i l i
l
i fi i
b
f
occurrences of the event are theoretically possible We are not restricted to a fixed
possible. We are not restricted to a fixed number of trials.
The events occur independently both within
The events occur independently both within the same interval and between consecutive intervals.
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The
The Poisson Distribution
Poisson Distribution

X
(0→∞) ◦ the number of occurrences of some event of interest over a given interval

λ =np
(constant)
◦ The average number of occurrences of the event in an interval

e =Natural logarithms ≈ 2.71828
g
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The
The Poisson Distribution
Poisson Distribution
Function of P(X),
mean, variance
i
Mean= np
Variance= np(1‐p) ≈ np
Mean=variance=np = λ
Try to use Try to use Table A.2 Table A.2 in Appendix A
in Appendix A
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The
The Poisson Distribution
Poisson Distribution

Ex: We are interested in the probability distribution of Ex: We are interested in the probability distribution of
the number of individuals involved in a motor vehicle accident each year (x)
accident
each year (x) in
in a town of the US.
a town of the US.
◦ p=0.00024, n=10,000 2010/3/17
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The
The Poisson Distribution
Poisson Distribution
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
Small λ → Poisson distribution is highly skewed

λ increase → the distribution becomes symmetric increase → the distribution becomes symmetric
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THE NORMAL DISTRIBUTION
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What is the Continuous Distribution
What is the Continuous
What is the 

In the binomial or Poisson distribution is for In
the binomial or Poisson distribution is for
discrete random variable ( x is integer value), not for continuous random variable
not for continuous random variable.
The probability density
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The Normal Distribution
(also called Gaussian Dist. or Bell
(also called Gaussian Dist. or Bell‐‐shaped Curve)



The most common continuous distribution
Symmetrical, mean=median=mode
y
,
The probability density
◦ μ (mu): mean
(mu): mean
◦ σ (sigma): standard deviation
◦ π (pi): a constant, 3.14159
(pi): a constant 3 14159
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The Normal Distribution
The Normal Distribution
μ =129
σ =19.8
19 8
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The Normal Distribution
The Normal Distribution
The total area under the Th
t t l
d th
curve must be 1
145 mm Hg
P(x=90.2)=0
P(X < 90.2)=0.025
P(90.2 < X < 167.8)=0.95
(
)
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
If
If ◦ p is constant but n approaches infinity in a binomial distribution
◦ λ approaches infinity in a Poisson distribution
→ Normal distribution shape
→ Normal distribution shape
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The Standard Normal Distribution
The Standard Normal Distribution



It s impossible to tabulate the area associated It’s
impossible to tabulate the area associated
with every normal distribution curve.
W
We can The standard normal distribution
◦ μ =0, σ =1
Use Table A.3 in Appendix A
Use Table A.3 Use Table A 3 in Appendix A
in Appendix A
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The
The Standard Normal Distribution
Standard Normal Distribution
Table A.3 in Appendix A
Table A.3 in Appendix A

For a particular value of z, the entry in the body of the table specifies the area beneath the
of the table specifies the area beneath the curve to the right of z, or P(Z>z)
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The
The Standard Normal Distribution
Standard Normal Distribution
Use table A.3 in Appendix A
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The
The Standard Normal Distribution
Standard Normal Distribution

If P(Z>z)=0 10 z=?
If P(Z>z)=0.10, z=?
Use table A.3 in Appendix A
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The
The Standard Normal Distribution
Standard Normal Distribution
Normal dist.
μ =2
σ =0.5
Standard normal dist.
Z score
By transforming X into Z, we can use the table A.3 (standard normal curve) to estimate probabilities associated with X
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The
The Standard Normal Distribution
Standard Normal Distribution
Let X be a random variable that represents systolic blood pressure For the population of 18 to 74‐year old males pressure . For the population of 18‐
to 74 year old males
in the US, systolic blood pressure is approximately normally distributed with mean 129 mm Hg and standard deviation 19 9 mm Hg
deviation 19.9 mm Hg.
μ =129
σ =19.8
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The
The Standard Normal Distribution
Standard Normal Distribution
μ =129
σ =19.8
P(90.2 < X < 167.8)= ?
(
)
When X=90.2→ Z=(90.2‐129)/19.8=‐1.96
(
)
X=167.8→ Z=(167.8‐129)/19.8= 1.96
P(‐1.96 < Z < 1.96)= 0.95 → P(90.2 < X < 167.8)= 0.95
(
)
→ (
)
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END
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