is a joint pdf. This definition can be extended easily to include more than two random variables.

Covariance between two random variables.

If x and y are random variables, then the covariance between the two variables,
$Cov\left(x,y\right)$ or
${\sigma}_{xy},$ is defined to be
$$Cov\left(x,y\right)=E\left[\left(x-{\mu}_{x}\right)\left(y-{\mu}_{y}\right)\right].$$ Expansion gives the alternative definition that
${\sigma}_{xy}=E\left(xy\right)-{\mu}_{x}{\mu}_{y}.$

Stochastic independence.

The random variables
x and
y are stochastically independent if and only if
${\sigma}_{xy}=0.$ An equivalent definition of independence is that
x and
y are stochastically independent if and only if
$$f\left(x,y\right)=f\left(x\right)f\left(y\right),$$ or, in words, if the joint pdf of the two random variables is equal to the product of the pdf of each random variable. From the definition of covariance it is easy to see that if two random variables are stochastically independent then
$E\left(xy\right)={\mu}_{x}{\mu}_{y}.$

Correlation coefficient.

The correlation coefficient,
$\rho ,$ is defined to be
$${\rho}_{xy}=\frac{{\sigma}_{xy}}{{\sigma}_{x}{\sigma}_{y}}.$$ The correlation coefficient is a unitless number that varies between -1 and +1. Clearly, two random variables are stochastically independent if and only if
${\rho}_{xy}=0.$

Discrete distributions

Binomial distribution.

The discrete random variable
x has a binomial distribution if
$$f\left(x\right)=\{\begin{array}{l}\left(\begin{array}{l}n\\ x\end{array}\right){p}^{x}{\left(1-p\right)}^{n-x},\text{}x=0,1,\dots ,n\\ 0\text{elsewhere}\end{array}$$ where
$\left(\begin{array}{l}n\\ x\end{array}\right)=\frac{n!}{x!\left(n-x\right)}.$ For the binomial distribution,
$\mu =np$ and
${\sigma}^{2}=np\left(1-p\right).$

Uniform distribution.

The discrete random variable
x has a uniform distribution if
$$f\left(x\right)=\left\{\begin{array}{l}\frac{1}{b-a}\text{if}a\le x\le b\\ 0\text{elsewhere}\end{array}\right\}.$$ The mean and variance of the uniform distribution are
$\mu =\frac{a+b}{2}$ and
${\sigma}^{2}=\frac{{\left(b-a\right)}^{2}}{12}.$

Poisson distribution.

The discrete random variable
x has a Poisson distribution if
$$f\left(x\right)=\{\begin{array}{l}\frac{{m}^{x}{e}^{-m}}{x!},\text{}x=0,1,\dots \\ 0\text{elsewhere}\end{array}$$ For the Poisson distribution
$\mu ={\sigma}^{2}=m.$ The Poisson distribution is used quite often in queuing theory to, among other things, describe the arrival of customers at a cashier's station.

Continuous distributions

Expotential distribution.

The continuous random variable
x has an exponential distribution if
$$f\left(x\right)=\left\{\begin{array}{l}\lambda {e}^{-\lambda x},\text{for}x\ge 0\text{}\\ \text{0for}x0\end{array}\right\}.$$ The cumulative exponential distribution is given by
$F\left(x\right)=1-{e}^{-\lambda x},$ for
$x\ge 0.$ The exponential distribution describes the times between events that occur continuously and independently at a constant rate (as in a Poisson process). The mean and variance of an exponential distribution are
$\mu ={\lambda}^{-1}$ and
${\sigma}^{2}={\lambda}^{-2}.$

Cauchy distribution.

A random variable
x , where
$-\infty <x<\infty ,$ has a Cauchy (or Cauchy-Lorentz) distribution if its pdf is
$$f\left(x\right)=\frac{1}{\pi}\left[\frac{\gamma}{{\left(x-{x}_{0}\right)}^{2}+{\gamma}^{2}}\right].$$ The parameter
${x}_{0}$ locates the peak of the pdf while
γ specifies the half-width of the pdf at the half maximum. Figure 3 shows the pdf and cumulative function for two values of these two parameters.

Normal distribution.

The continuous random variable
x has a normal distribution with a mean of
$\mu $ and a variance of
${\sigma}^{2}$ if its pdf is
$$f\left(x\right)=\frac{1}{\sigma \sqrt{2\pi}}{e}^{-\frac{{\left(x-\mu \right)}^{2}}{2{\sigma}^{2}}}$$ for
$-\infty \le x\le \infty .$ The distribution is symmetric around the mean.

Questions & Answers

I only see partial conversation and what's the question here!

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

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?