1. Lecture 5 Section 1.6 Objectives:
Several Useful Discrete Distributions
Binomial Distribution
Poisson Distribution
Poisson Approximation to the Binomial

2. Discrete Distribution
A distribution for a discrete variable x is specified by a mass function p(x) satisfying,
p(x) ≥ 0 for all x b.
Two discrete distributions that appear most frequently in statistical applications will be introduced.

3. The Binomial Distribution
Some experiments can be viewed as a sequence of independent Bernoulli trials, where each outcome is a “success” or “failure”. The total number of “successes” from such an experiment is often of interest rather than the individual outcomes.
Suppose that there is a fixed number n of trials that are independent and each trial has the same proportion π of success. Let x=the number of success in n independent trials with proportion of success π. Then, the mass function of x is given by
Note that x~B(n,π) means that the variable x has a binomial distribution with number of trials n and proportion of success π.

4. Some Examples
Coin Toss. Toss a coin three times. Let x - number of heads.
Find the distribution for x.
b. Find the proportion of getting two heads.
Female Chicks. Leghorn chickens are raised for laying eggs. Let π = 0.5 be the proportion of a female chick hatching. Assuming independence, let x - # of female chicks out of 10 newly hatched chicks selected at random.
b. Find the proportion of 5 or fewer female chicks.

5. The Poisson Distribution
The Poisson distribution is used as a model for the count of random events occurring over time, e.g., the number of phone calls arriving at a switchboard between 9 and 10am, the number of jobs arriving at a work station or the number of radioactive particle emissions.
The Poisson mass function is
where the parameter λ must satisfy λ > 0.
Note that x ~ Poisson(λ) means that the variable x has a Poisson distribution with parameter λ.

6. Example 1.21
Let x denote the number of creatures of a particular type captured in a trap during
a given time period. Suppose that x has a Poisson distribution with 4.5, so on
average, the traps will contain 4.5 creatures. [The article “Dispersal Dynamics of
the Bivalve Gemma Gemma in a Patchy Environment” (Ecological Monographs,
1995: 1-20) suggests this model; the bivalve Gemma gemma is a small clam].
Find the proportions of traps with five creatures.
Find the proportion of traps with at most five creatures.
Find the proportion of traps with at least six creatures.

7. The Poisson Approximation to the Binomial Distribution
The Poisson distribution is a limiting form of the binomial distribution:
with λ = nπ as n→∞ and π→0 in such a way that nπ approaches a positive constant λ.
So, the Poisson distribution can be used to model the number of occurrences of a rare event when the number of opportunities for the event is very large, but the probability that the event occurs in any specific instance is very small. Moreover, the occurrences are independent Bernoulli trials with the same proportion of success. For example, the number of flaws in 100 feet of wire, or the number of defects in a 100-foot roll of aluminum screen that is 2 feet wide, the number of accidents at an intersection, the number of earthquakes, the number of leukemia cases, etc.

8. Example
A manufacturer of Christmas tree light bulbs knows that 2% of its bulbs are defective. Assuming independence, we have a binomial distribution with parameters π=.02 and n=100. Find the proportion that a box of 100 of these bulbs contains at most three defective bulbs. Let x be the number of defective bulbs.
Binomial distribution, B(100,0.02) (gray bars) and Poisson distribution, Poisson(2) (black bars).