# 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 $\sigma =\sqrt{\mu }$

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
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