Introduction to Biostatistics (PUBHLTH 540) Lecture 7: Binomial and Poisson Distributions

1
Introduction to Biostatistics(PUBHLTH 540)
Lecture 7 Binomial and Poisson Distributions
Acknowledgement Thanks to Professor Pagano
(Harvard School of Public Health) for lecture
material
2
All exact science is dominated by the idea of
approximation. Bertrand Russell (1872-1970)
3
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

4
Random Variables

• Random Variable is a function that assigns
• specific numerical values to all possible
• outcomes of experiment
• Probability distributions are associated with
  random variables to describe the probabilities of
  the various outcomes of an experiment

1,2,3,4,5,6















5
Random Variables

• Types
• Discrete Bernoulli, Binomial, Poisson
• Continuous Exponential, Normal

6
Random Variables

• Bernoulli
• Binomial
• Poisson

7
Random Variables - Bernoulli
When outcomes of experiment are binary
Dichotomous (Bernoulli) X 0 or 1
P(X1) p
P(X0) 1-p
e.g. Heads, Tails True, False
Success, Failure
8
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)

9
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?

10
e.g.
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 (vaccinated person) 0 ? failure
(unvaccinated person)
1 Trial P(0) 1-p 0.2 P(1) p 0.8
11
e.g. 2 trials
Binomial - Example
2 Trials
Trials Probability 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
12
e.g. continued
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(X0) (1-p)2 0.04 P(X1) 2p(1-p)
0.32 P(X2) p2 0.64
13
Factorial notation
Binomial Coefficient
So,
14
Binomial Coefficient
Binomial Coefficient
By convention 0! 1
15
Binomial Distribution
X number of successes in n trials
Parameters p probability of success n
number of trials
16
Binomial Distribution
N2 trials Xnum. successes P(X0) (1-p)2
0.04 P(X1) 2p(1-p) 0.32 P(X2) p2
0.64
17
Binomial with n10 and p0.5
Binomial Distribution
18
Binomial with n10 and p0.29
Binomial Distribution
19
Mean and variance
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)
20
Binomial Distribution - Moments
e.g. p0.5 n10 Mean np 10 0.5
5 Variance np(1-p) 10(0.5)(0.5) 2.5
21
Poisson Distribution
Xnumber 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 ?.

22
Poisson Distribution
Source http//en.wikipedia.org/wiki/Poisson_distr
ibution
23
Poisson Distribution
For the Poisson one parameter
?
Poisson
Binomial
np np(1-p) ? np when p is small
Mean ? Variance ?
24
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

25
Poisson with ? 2.4
Poisson Distribution