Transcript Document
Introduction to
Biostatistics
(PUBHLTH 540)
Lecture 7:
Binomial and Poisson
Distributions
Acknowledgement: Thanks to Professor Pagano
(Harvard School of Public Health) for lecture material 1
All exact science is dominated
by the idea of approximation.
Bertrand Russell (1872-1970)
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Random Variables
Variable: measurable characteristic
Random Variable: variable that can
have different outcomes of an
experiment, determined by chance
Examples:
•X = outcome of roll of a die,
•Y = outcome of a coin toss,
•Z = height
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Random Variables
• Random Variable is a function that assigns
specific numerical values to all possible
outcomes of experiment
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{1,2,3,4,5,6}
• Probability distributions are associated with random
variables to describe the probabilities of the various
outcomes of an experiment
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Random Variables
• Types:
– Discrete: Bernoulli, Binomial, Poisson
– Continuous: Exponential, Normal
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Random Variables
• Bernoulli
• Binomial
• Poisson
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Random Variables - Bernoulli
When outcomes of experiment are binary
Dichotomous (Bernoulli): X = 0 or 1
P(X=1) = p
P(X=0) = 1-p
e.g. Heads, Tails
True, False
Success, Failure
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Binomial Distribution
•A sequence of independent
Bernoulli trials (n) with constant
probability of success at each
trial (p) and we are interested in
the total number of successes (x).
•Assumptions:
•N trials of an experiment
•Each experiment results in one of 2
outcomes (binary)
•Each trial is independent of the other
trials
•In each trial, the probability of ‘success’
is constant (p)
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Binomial - Examples
Can the binomial distribution be used in
the settings below?
• 10 tosses of a coin – Yes/No?
• 10 rolls of a die – Yes/No?
• 10 rolls of a die and the number time it turns up a
6 – Yes/No?
• Number of individuals who have a particular
disease in a town – Yes/No?
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Binomial - Example
Suppose that 80% of the villagers
should be vaccinated. What is the
probability that at random you choose
a vaccinated villager?
1 success
0 failure
(vaccinated person)
(unvaccinated person)
1 Trial
P(0) = 1-p = 0.2
P(1) = p = 0.8
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Binomial - Example
2 Trials:
Trials
Probability
#succ. Prob
(0,0)
(1-p) (1-p)
0
0.04
(0,1)
(1-p) p
1
0.16
(1,0)
p (1-p)
1
0.16
(1,1)
p p
2
0.64
P(0 vaccinated) = (1-p)2
P(1 vaccinated) = 2p(1-p)
P(2 vaccinated) = p2
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Binomial - Example
Experiment: Sample two villagers
at random and determine whether
they are vaccinated
X number of successes
n = 2, the number of trials
P(X=0) = (1-p)2 = 0.04
P(X=1) = 2p(1-p) = 0.32
P(X=2) = p2
= 0.64
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Binomial Coefficient
Factorial notation:
1 ´ 2 = 2!
1 ´ 2´ 3 = 3!
1 ´ 2´ 3´ 4 = 4!
M
1 ´ 2´ 3´ K ( n - 1)´ n = n !
So,
3! = 6,
4! = 24,
5! = 120
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Binomial Coefficient
By convention: 0! = 1
Binomial Coefficient:
n!
n =
x
x ! ( n - x )!
()
n = 1, 2, K
x = 0,1, K , n
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Binomial Distribution
X = number of successes in n trials
Parameters:
p = probability of success
n = number of trials
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Binomial Distribution
N=2 trials; X=num. successes
P(X=0) = (1-p)2 = 0.04
P(X=1) = 2p(1-p) = 0.32
P(X=2) = p2
= 0.64
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Binomial Distribution
Binomial with n=10 and p=0.5
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Binomial Distribution
Binomial with n=10 and p=0.29
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Binomial Distribution - Moments
For X ~ Binomial(n,p)
(i.e. n = Num. Trials,
p = Probability of success in each trial)
Then
Mean = E(X) = np
Variance = Var(X) = np(1-p)
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Binomial Distribution - Moments
e.g. p=0.5 n=10
Mean = np = 10
0.5 = 5
Variance =np(1-p) = 10(0.5)(0.5) =2.5
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Poisson Distribution
X=number of occurrences of event in a given
time period
1. The probability an event occurs in
the interval is proportional to the
length of the interval.
2. An infinite number of occurrences
are possible.
3. Events occur independently at a
rate .
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Poisson Distribution
Source: http://en.wikipedia.org/wiki/Poisson_distribution
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Poisson Distribution
For the Poisson
one parameter:
Poisson
Binomial
Mean =
np
Variance =
np(1-p)
np
when p
is small
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Poisson Distribution - Example
e.g. Probability of an accident in
a year is 0.00024. So in a town
of 10,000, the rate
= np
= 10,000 x 0.00024 = 2.4
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Poisson Distribution
Poisson with =2.4
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