# 4.6 Summary of functions

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This module provides a review of the binomial, geometric, hypergeometric, and Poisson probability distribution functions and their properties.
Formula

## Binomial

$X$ ~ $B\left(n,p\right)$

$X$ = the number of successes in $n$ independent trials

$n$ = the number of independent trials

$X$ takes on the values $x=$ 0,1, 2, 3, ..., $n$

$p$ = the probability of a success for any trial

$q$ = the probability of a failure for any trial

$p+q=1\phantom{\rule{15pt}{0ex}}q=1-p$

The mean is $\mu =\mathrm{np}$ . The standard deviation is $\sigma =\sqrt{\mathrm{npq}}$ .

Formula

## Geometric

$X$ ~ $G\left(p\right)$

$X$ = the number of independent trials until the first success (count the failures and the first success)

$X$ takes on the values $x$ = 1, 2, 3, ...

$p$ = the probability of a success for any trial

$q$ = the probability of a failure for any trial

$p+q=1$

$q=1-p$

The mean is $\mu =\frac{1}{p}$

Τhe standard deviation is $\sigma =\sqrt{\frac{1}{p}\left(\left(\frac{1}{p}\right)-1\right)}$

Formula

## Hypergeometric

$X$ ~ $H\left(r,b,n\right)$

$X$ = the number of items from the group of interest that are in the chosen sample.

$X$ may take on the values $x$ = 0, 1, ..., up to the size of the group of interest. (The minimum valuefor $X$ may be larger than 0 in some instances.)

$r$ = the size of the group of interest (first group)

$b$ = the size of the second group

$n$ = the size of the chosen sample.

$(n, r+b)$

The mean is: $\mu =\frac{nr}{r+b}$

The standard deviation is: $(\sigma , \sqrt{\frac{rbn\left(r+b-n\right)}{{\left(r+b\right)}^{2}\left(r+b-1\right)}})$

Formula

## Poisson

$X$ ~ $\text{P(μ)}$

$X$ = the number of occurrences in the interval of interest

$X$ takes on the values $x$ = 0, 1, 2, 3, ...

The mean $\mu$ is typically given. ( $\lambda$ is often used as the mean instead of $\mu$ .) When the Poisson is used to approximate the binomial, we use the binomialmean $\mu =np$ . $n$ is the binomial number of trials. $p$ = the probability of a success for each trial. This formula is valid when n is "large" and $p$ "small" (a general rule is that $n$ should be greater than or equal to $20$ and $p$ should be less than or equal to $0.05$ ). If $n$ is large enough and $p$ is small enough then the Poisson approximates the binomial very well. The variance is ${\sigma }^{2}=\mu$ and the standard deviation is $\sigma =\sqrt{\mu }$

#### Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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
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Source:  OpenStax, Quantitative information analysis iii. OpenStax CNX. Dec 25, 2009 Download for free at http://cnx.org/content/col11155/1.1
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