So, the probability goes to zero at a rate of at least
${n}^{-1}$ .
However, it turns out that this is an extremely loose bound. Accordingto the Central Limit Theorem
Chernoff's bound is based on finding the value of
$s$ that
minimizes the upper bound. If
$Z$ is a sum of independent random
variables. For example, say
Thus, the problem of finding a tight bound boils down to finding a
good bound for
$E\left[{s}^{s({L}_{i}-E\left[{L}_{i}\right])}\right]$ . Chernoff ('52), first
studied this situation for binary random variables. Then,Hoeffding ('63) derived a more general result for arbitrary
bounded random variables.
Hoeffding's indequality
Theorem
Hoeffding's inequality
Let
${Z}_{1},{Z}_{2},...,Zn$ be independent bounded random variables such that
${Z}_{i}\in [{a}_{i},{b}_{i}]$ with probability 1. Let
${S}_{n}={\sum}_{i=1}^{n}{Z}_{i}$ . Then for any
$t>0$ , we have
Similarly,
$P(E\left[{S}_{n}\right]-{S}_{n}\ge t)\le {e}^{\frac{-2{t}^{2}}{{\sum}_{i=1}^{n}{({b}_{i}-{a}_{i})}^{2}}}$ .
This completes the proof of the Hoeffding's theorem.
Application
Let
${Z}_{i}={1}_{f\left({X}_{i}\right)\ne {Y}_{i}}-R\left(f\right),$ as in the
classification problem. Then for a fixed f, it follows fromHoeffding's inequality (i.e., Chernoff's bound in this special case)
that
Now, we want a bound like this to hold uniformly for all
$f\in \mathcal{F}$ .
Assume that
$\mathcal{F}$ is a finite collection of models and let
$\left|\mathcal{F}\right|$ denote its cardinality. We would like to bound the
probability that
${max}_{f\in \mathcal{F}}|\widehat{{R}_{n}}\left(f\right)-R\left(f\right)|\ge \u03f5$ . Note that the event
In this morden time nanotechnology used in many field .
1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc
2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc
3- Atomobile -MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.