1. Share one real-world binomial distribution situation and one real-world Poisson distribution
situation. Be sure to explain why each example is defined as binomial or Poisson. How would
you characterize the difference between the two types of distributions?
Solution
1. Here%u2019s an example where the Poisson distribution was used in a maternity hospital to
work out how many births would be expected during the night.
The hospital had 3000 deliveries each year, so if these happened randomly around the clock 1000
deliveries would be expected between the hours of midnight and 8.00 a.m. This is the time when
many staff are off duty and it is important to ensure that there will be enough people to cope with
the workload on any particular night.
The average number of deliveries per night is 1000/365, which is 2.74. From this average rate
the probability of delivering 0,1,2, etc babies each night can be calculated using the Poisson
distribution. Some probabilities are:
P(0) 2.740 e-2.74 / 0! = 0.065
P(1) 2.741 e-2.74 / 1! = 0.177
P(2) 2.742 e-2.74 / 2! = 0.242
P(3) 2.743 e-2.74 / 3! = 0.221
It is very important in measuring statistics. A binomial distribution is the collection of
probabilities of an event with a given probability happening x times given a number of trials.
Say you roll a 6-sided die 60 times and you count the number of time it comes up 6 (or any
number 1 to 6, doesn't matter). The most likely outcome is that you will count 10 instances of
sixes coming up, but there is also a pretty good chance you'll count 9 or 11. It would be very
unlikely that you'd count 2 sixes, however, and the most extreme case, 60 sixes, would be
almost unfathomably unlikely (but possible). Since this is a simple, clean problem we could
figure out the probabilities mathematically. We could also add up the probabilities of a range of
possibilities, say 7 to 13 instances, and come up with the odds of a 6-sided die tossed 60 times
coming up six between 7 and 13 times.
That last example is the basis of polling techniques. What we are looking for from polls is the
true likelihood that a randomly chosen person would answer a question in a certain way. That is
our probability, p, and it is the unknown. What pollsters do, then, is try their best to select a

2. representative sample and start asking the question (rolling the die). They might ask 1000 people
and record their responses. But before they start asking the question, they've already figured out
what the probability range is going to look like with that many respondents. The "p" is still
theoretical, but they can figure out their margin of error mathematically.
the Poisson and the binomial are discrete random variables, the normal is a continuous random
variable.
Notes about the Binomial:
In general, if X has the binomial distribution with n trials and a success probability of p then
P[X = x] = n!/(x!(n-x)!) * p^x * (1-p)^(n-x)
for values of x = 0, 1, 2, ..., n
P[X = x] = 0 for any other value of x.
The probability mass function is derived by looking at the number of combination of x objects
chosen from n objects and then a total of x success and n - x failures.
Or, in other words, the binomial is the sum of n independent and identically distributed Bernoulli
trials.
the Poisson:
The Poisson distribution can be derived from the binomial distribution. The Poisson is nothing
more than the limiting case of the Binomial where n is large and p is small.
A good way to identify when you need to use the Poisson distribution is when the problem
requires you to use a rate. This is not always true, but more often than not remembering this will
help you to identify a Poisson model.
It is great for models such as yeast cell distribution in stock solution, (first use in practice by
W.S. Gosset at the Guinness brewery). It's great for radio active decay... etc.
X ~ Poisson( %u03BBt )
P(X = x) = ( %u03BBt )^x * exp( -%u03BBt ) / x! for x = 0, 1, 2, 3, 4, ...
P(X = x) = 0 otherwise
the mean of the Poisson distribution is the parameter, %u03BBt

3. the variance of the Poisson distribution is the parameter, %u03BBt