# 4.4 Poisson distribution  (Page 3/18)

 Page 3 / 18

On May 13, 2013, starting at 4:30 PM, the probability of low seismic activity for the next 48 hours in Alaska was reported as about 1.02%. Use this information for the next 200 days to find the probability that there will be low seismic activity in ten of the next 200 days. Use both the binomial and Poisson distributions to calculate the probabilities. Are they close?

Let X = the number of days with low seismic activity.

Using the binomial distribution:

• $P\left(x=10\right)=\frac{200!}{10!\left(200-10\right)!}×{.0102}^{10}=0.000039$

Using the Poisson distribution:

• Calculate μ = np = 200(0.0102) ≈ 2.04
• $P\left(x=10\right)=\frac{{\mu }^{x}{e}^{\mathrm{-\mu }}}{\mathrm{x!}}=\frac{{2.04}^{10}{e}^{-2.04}}{10!}=0.000045$

We expect the approximation to be good because n is large (greater than 20) and p is small (less than 0.05). The results are close—both probabilities reported are almost 0.

## Estimating the binomial distribution with the poisson distribution

We found before that the binomial distribution provided an approximation for the hypergeometric distribution. Now we find that the Poisson distribution can provide an approximation for the binomial. We say that the binomial distribution approaches the Poisson. The binomial distribution approaches the Poisson distribution is as n gets larger and p is small such that np becomes a constant value. There are several rules of thumb for when one can say they will use a Poisson to estimate a binomial. One suggests that np , the mean of the binomial, should be less than 25. Another author suggests that it should be less than 7. And another, noting that the mean and variance of the Poisson are both the same, suggests that np and npq , the mean and variance of the binomial, should be greater than 5. There is no one broadly accepted rule of thumb for when one can use the Poisson to estimate the binomial.

As we move through these probability distributions we are getting to more sophisticated distributions that, in a sense, contain the less sophisticated distributions within them. This proposition has been proven by mathematicians. This gets us to the highest level of sophistication in the next probability distribution which can be used as an approximation to all of those that we have discussed so far. This is the normal distribution.

A survey of 500 seniors in the Price Business School yields the following information. 75% go straight to work after graduation. 15% go on to work on their MBA. 9% stay to get a minor in another program. 1% go on to get a Master's in Finance.

What is the probability that more than 2 seniors go to graduate school for their Master's in finance?

This is clearly a binomial probability distribution problem. The choices are binary when we define the results as "Graduate School in Finance" versus "all other options." The random variable is discrete, and the events are, we could assume, independent. Solving as a binomial problem, we have:

Binomial Solution

$n*p=500*0.01=5=µ$
$P\left(0\right)=\frac{500!}{0!\left(500-0\right)!}{0.01}^{0}\left(1-0.01{\right)}^{{500}^{{-}^{0}}}=0.00657$
$P\left(1\right)=\frac{500!}{1!\left(500-1\right)!}{0.01}^{1}\left(1-0.01{\right)}^{{500}^{{-}^{1}}}=0.03318$
$P\left(2\right)=\frac{500!}{2!\left(500-2\right)!}{0.01}^{2}\left(1-0.01{\right)}^{{500}^{{-}^{2}}}=0.08363$

Adding all 3 together = 0.12339

$1-0.12339=0.87661$

Poisson approximation

$n*p=500*0.01=5=\mu$
$n*p*\left(1-p\right)=500*0.01*\left(0.99\right)\approx 5={\sigma }^{2}=\mu$
$P\left(X\right)=\frac{{e}^{{\mathrm{-np}}^{}}\left(np{\right)}^{x}}{x!}=\left\{P\left(0\right)=\frac{{e}^{-5}*{5}^{0}}{0!}\right\}+\left\{P\left(1\right)=\frac{{e}^{-5}*{5}^{1}}{1!}\right\}+\left\{P\left(2\right)=\frac{{e}^{-5}*{5}^{2}}{2!}\right\}$
$0.0067+0.0337+0.0842=0.1247$
$1-0.1247=0.8753$

An approximation that is off by 1 one thousandth is certainly an acceptable approximation.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Got questions? Join the online conversation and get instant answers!

#### Get Jobilize Job Search Mobile App in your pocket Now! By Sean WiffleBoy By Kevin Amaratunga By OpenStax By Mistry Bhavesh By OpenStax By Frank Levy By Sarah Warren By Jemekia Weeden By OpenStax By Rebecca Butterfield