# 4.7 Poisson

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This module describes the characteristics of a Poisson experiment and the Poisson probability distribution. This module is included in the Elementary Statistics textbook/collection as an optional lesson.

Characteristics of a Poisson experiment:

1. The Poisson gives the probability of a number of events occurring in a fixed interval of time or space if these events happen with a known average rate and independently of the time since the last event. Forexample, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on the average, there are 5 words spelled incorrectly in100 pages. The interval is the 100 pages.
2. The Poisson may be used to approximate the binomial if the probability of success is "small" (such as 0.01) and the number of trials is "large" (such as 1000). You will verify therelationship in the homework exercises. $n$ is the number of trials and $p$ is the probability of a "success."
Poisson probability distribution . The random variable $X=$ the number of occurrences in the interval of interest. The mean and variance are given in the summary.

The average number of loaves of bread put on a shelf in a bakery in a half-hour period is 12. Of interest is the number of loaves of bread put on the shelf in 5 minutes. Thetime interval of interest is 5 minutes. What is the probability that the number of loaves, selected randomly, put on the shelf in 5 minutes is 3?

Let $X$ = the number of loaves of bread put on the shelf in 5 minutes. If the average number of loaves put on the shelf in 30 minutes (half-hour) is 12, then the average number of loaves put on the shelf in 5 minutes is

$(\left(\frac{5}{30}\right)\cdot 12, 2)$ loaves of bread

The probability question asks you to find $\mathrm{P\left(x = 3\right)}$ .

A certain bank expects to receive 6 bad checks per day, on average. What is the probability of the bank getting fewer than 5 bad checks on any given day? Of interestis the number of checks the bank receives in 1 day, so the time interval of interest is 1 day. Let $X$ = the number of bad checks the bank receives in one day. If the bank expects to receive 6 bad checks per day then the average is 6 checks per day.The probability question asks for $(P\left(x, 5\right))$ .

You notice that a news reporter says "uh", on average, 2 times per broadcast. What is the probability that the news reporter says "uh" more than 2 times per broadcast.

This is a Poisson problem because you are interested in knowing the number of times the news reporter says "uh" during a broadcast.

What is the interval of interest?

What is the average number of times the news reporter says "uh" during one broadcast?

2

Let $X$ = ____________. What values does $X$ take on?

Let $X$ = the number of times the news reporter says "uh" during one broadcast .
$x$ = 0, 1, 2, 3, ...

The probability question is $\text{P(______)}$ .

$\text{P(x>2)}$

## Notation for the poisson: p = poisson probability distribution function

$X$ ~ $\text{P(μ)}$

Read this as " $X$ is a random variable with a Poisson distribution." The parameter is $\mu$ (or $\lambda$ ). $\mu$ (or $\lambda$ ) = the mean for the interval of interest.

Leah's answering machine receives about 6 telephone calls between 8 a.m. and 10 a.m. What is the probability that Leah receives more than 1 call in the next 15 minutes?

Let $X$ = the number of calls Leah receives in 15 minutes. (The interval of interest is 15 minutes or $\frac{1}{4}$ hour.)

$x$ = 0, 1, 2, 3, ...

If Leah receives, on the average, 6 telephone calls in 2 hours, and there are eight 15 minutes intervals in 2 hours, then Leah receives

$(\frac{1}{8}\cdot 6, 0.75)$

calls in 15 minutes, on the average. So, $\mu$ = 0.75 for this problem.

$X$ ~ $\text{P(0.75)}$

Find $(P\left(x, 1\right))$ . $((P\left(x, 1\right)), 0.1734)$ (calculator or computer)

TI-83+ and TI-84: For a general discussion, see this example (Binomial) . The syntax is similar. The Poisson parameter list is ( $\mu$ for the interval of interest, number). For this problem:

Press 1- and then press 2nd DISTR. Arrow down to C:poissoncdf. Press ENTER. Enter .75,1). The result is $((P\left(x, 1\right)), 0.1734)$ . NOTE: The TI calculators use $\lambda$ (lambda) for the mean.

The probability that Leah receives more than 1 telephone call in the next fifteen minutes is about 0.1734.

The graph of $X$ ~ $\text{P(0.75)}$ is:

The y-axis contains the probability of $x$ where $X$ = the number of calls in 15 minutes.

where we get a research paper on Nano chemistry....?
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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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1 It is estimated that 30% of all drivers have some kind of medical aid in South Africa. What is the probability that in a sample of 10 drivers: 3.1.1 Exactly 4 will have a medical aid. (8) 3.1.2 At least 2 will have a medical aid. (8) 3.1.3 More than 9 will have a medical aid.