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 ( )

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 • • • • • • {1,2,3,4,5,6} • • • • • • • • • • • • • •
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(X=1) = p
P(X=0) = 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. Can the binomial distribution be used in
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. Binomial - Example 1 Trial P(0) = 1-p = 0.2 P(1) = p = 0.8
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. Binomial - Example 2 Trials: P(0 vaccinated) = (1-p)2
e.g. 2 trials
Binomial - Example
2 Trials:
Trials
Probability
#succ.
Prob
(0,0)
(1-p)
0.04
(0,1) p 1
0.16
(1,0)
(1,1) 2
0.64
P(0 vaccinated) = (1-p)2
P(1 vaccinated) = 2p(1-p)
P(2 vaccinated) = p2

12. Binomial - Example X  number of successes
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(X=0) = (1-p)2 = 0.04
P(X=1) = 2p(1-p) = 0.32
P(X=2) = p = 0.64

13. Binomial Coefficient
Factorial notation:
So,

14. Binomial Coefficient:
By convention: 0! = 1
Binomial Coefficient:

15. Binomial Distribution
X = number of successes in n trials
Parameters:
p = probability of success
n = number of trials

16. N=2 trials; X=num. successes
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) = p = 0.64

17. Binomial Distribution
Binomial with n=10 and p=0.5

18. Binomial Distribution
Binomial with n=10 and p=0.29

19. Binomial Distribution - Moments
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. p=0.5 n=10
Mean = np = = 5
Variance =np(1-p) = 10(0.5)(0.5) =2.5

21. Poisson Distribution
X=number of occurrences of event in a given time period
The probability an event occurs in the interval is proportional to the length of the interval.
An infinite number of occurrences are possible.
Events occur independently at a rate .

22. Poisson Distribution
Source:

23. Poisson Distribution Mean =  For the Poisson one parameter:  np
Binomial
Mean = 
Variance =  np
np(1-p)
 np
when p
is small

24. Poisson Distribution - Example
e.g. Probability of an accident in a year is So in a town of 10,000, the rate
= np
= 10,000 x = 2.4

25. Poisson Distribution
Poisson with  =2.4