1. The document discusses binomial and Poisson distributions as types of random variables. Binomial distributions model experiments with a fixed number of trials with two possible outcomes, while Poisson distributions model rare, independent events occurring continuously over time or space. 2. Key properties of binomial distributions include the number of trials n, probability of success p, and outcomes being the number of successes. Mean is np and variance is np(1-p). Poisson distributions have a single parameter λ representing the expected number of events. 3. Examples are provided to illustrate calculating probabilities for binomial experiments and defining the Poisson parameter λ based on expected events per unit of time or space.

1. 1
Binomial and Poisson
Distributions

2. 2
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
Random Variables

3. 3
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
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{1,2,3,4,5,6}

4. 4
Random Variables
• Types:
– Discrete: Bernoulli, Binomial, Poisson
– Continuous: Exponential, Normal

5. 5
Random Variables
• Bernoulli
• Binomial
• Poisson

6. 6
Dichotomous (Bernoulli): X = 0 or 1
P(X=1) = p
P(X=0) = 1-p
e.g. Heads, Tails
True, False
Success, Failure
Random Variables - Bernoulli
When outcomes of experiment are binary

7. 7
•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)
Binomial Distribution

8. 8
Binomial - Examples
• 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?
Can the binomial distribution be used in
the settings below?

9. 9
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
Binomial - Example

10. 10
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
Binomial - Example

11. 11
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
Binomial - Example
Experiment: Sample two villagers
at random and determine whether
they are vaccinated

12. 12
3! 6, 4! 24, 5! 120= = =
So,
Factorial notation:
Binomial Coefficient

13. 13
( ) !
! ( )!
nn
x x n x
=
-
By convention: 0! = 1
1,2,
0,1, ,
n
x n
=
=
K
K
Binomial Coefficient:
Binomial Coefficient

14. 14
X = number of successes in n trials
Parameters:
p = probability of success
n = number of trials
Binomial Distribution

15. 15
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
Binomial Distribution

16. 16
Binomial with n=10 and p=0.5
Binomial Distribution

17. 17
Binomial with n=10 and p=0.29
Binomial Distribution

18. 18
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)
Binomial Distribution - Moments

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

20. 20
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 .
Poisson Distribution
X=number of occurrences of event in a given
time period

21. 21
Poisson Distribution
Source: http://en.wikipedia.org/wiki/Poisson_distribution

22. 22
For the Poisson
one parameter:
Mean =
Variance =
np
np(1-p)
np
when p
is small
Poisson Distribution
Poisson Binomial

23. 23
l= np
= 10,000 x 0.00024 = 2.4
e.g. Probability of an accident in
a year is 0.00024. So in a town
of 10,000, the rate
Poisson Distribution - Example

24. 24
Poisson with =2.4
Poisson Distribution