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

Is there any normative that regulates the use of silver nanoparticles?

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

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?