# 1.6 Basic definitions in stochastic processes

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A random or stochastic process is the assignment of a function of a real variable to each sample point  in a sample space. Thus, the process $X(, t)$ can be considered a function of two variables. For each  , the time function must be well-behaved and may or may not look random to the eye.Each time function of the process is called a sample function and must be defined over the entire domain of interest. For each $t$ , we have a function of  , which is precisely the definition of a random variable. Hence the amplitude of a random process is a random variable. The amplitude distribution of a process refers to the probability density function of the amplitude: $p(X(t), x)$ . By examining the process's amplitude at several instants, the joint amplitude distribution can also be defined.For the purposes of this module, a process is said to be stationary when the joint amplitude distribution depends on the differences between the selected time instants.

The expected value or mean of a process is the expected value of the amplitude at each $t$ . $(X(t))={m}_{X}(t)=\int_{()} \,d x$ x p X t x For the most part, we take the mean to be zero. The correlation function is the first-order joint moment between the process's amplitudes at two times. ${R}_{X}({t}_{1}, {t}_{2})=\int_{()} \,d {x}_{2}$ x 1 x 1 x 2 p X t 1 X t 2 x 1 x 2 Since the joint distribution for stationary processes depends only on the time difference, correlation functions of stationaryprocesses depend only on $\left|{t}_{1}-{t}_{2}\right|$ . In this case, correlation functions are really functions of a single variable (the time difference) and areusually written as ${R}_{X}()$ where $={t}_{1}-{t}_{2}$ . Related to the correlation function is the covariance function ${K}_{X}()$ , which equals the correlation function minus the square of the mean. ${K}_{X}()={R}_{X}()-{m}_{X}^{2}$ The variance of the process equals the covariance function evaluated as the origin. The power spectrum of a stationary process is the Fourier Transform of the correlationfunction. ${}_{X}()=\int_{()} \,d$ R X A particularly important example of a random process is white noise . The process $X(t)$ is said to be white if it has zero mean and a correlation function proportional to an impulse. $(X(t))=0$ ${R}_{X}()=\frac{{N}_{0}}{2}()$ The power spectrum of white noise is constant for all frequencies, equaling $\frac{{N}_{0}}{2}$ which is known as the spectral height .

The curious reader can track down why the spectral height of white noise has the fraction one-half init. This definition is the convention.

When a stationary process $X(t)$ is passed through a stable linear, time-invariant filter, the resulting output $Y(t)$ is also a stationary process having power density spectrum ${}_{Y}()=\left|H(i)\right|^{2}{}_{X}()$ where $H(i)$ is the filter's transfer function.

so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
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
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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