# Statistical terminology  (Page 4/4)

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## Log normal distribution.

The continuous random variable x has log normal distribution if y has a normal distribution and $x={e}^{y}.$ Thus, if $y\sim N\left(\mu ,{\sigma }^{2}\right),$ then the pdf of a log normal distribution is The mean and variance of x are ${\mu }_{x}={e}^{\mu +\frac{{\sigma }^{2}}{2}}$ and ${\sigma }_{x}^{2}=\left({e}^{{\sigma }^{2}}-1\right){e}^{2\mu +{\sigma }^{2}}.$ Because the distribution is skewed downward for variances over 1, the log normal distribution is sometimes used to describe income distributions (where there are relatively few very wealthy people and incomes generally are positive. Figure 4 shows the graphs of the pdf and cumulative functions for the log normal distributions for two values of σ .

## Gamma distribution.

A positive random variable x has a gamma distribution if its pdf is $f\left(x\right)=\frac{1}{\Gamma \left(\alpha \right){\beta }^{\alpha }}{x}^{\alpha -1}{e}^{-\frac{x}{\beta }}$ for $x>0$ and 0 elsewhere. $\Gamma \left(\alpha \right)$ is known as the gamma function and is defined to be $\Gamma \left(\alpha \right)={\int }_{0}^{\infty }{y}^{\alpha -1}{e}^{-y}dy=\left(\alpha -1\right)!.$ The gamma function is often used to model waiting times like waiting for death. Its mean and variance are given by $\mu =\alpha \beta$ and ${\sigma }^{2}=\alpha {\beta }^{2}.$

## Chi-square distribution.

A chi-square distribution ( ${\chi }^{2}\left(k\right)$ ) is the sum of k independent standard normal random variables and is a special case of the gamma distribution (with $\alpha =\frac{k}{2}$ and $\beta =2$ ). The pdf of a chi-square distribution with k degrees of freedom is $f\left(x\right)=\frac{1}{{2}^{k}{2}}\Gamma \left(\frac{k}{2}\right)}{x}^{\frac{k}{2}-1}{e}^{-\frac{x}{2}}$ where x >0. Its mean and variance are $\mu =k$ and ${\sigma }^{2}=2k.$ If $y=\sum _{i=1}^{k}{x}_{i}^{2}$ where the x i 's are independently drawn from the standard normal distribution (N(1, 0)), then ${y}_{i}\sim {\chi }^{2}\left(k\right).$

## Student's t-distribution.

Consider two random variables, x and v . Assume that $x\sim N\left(0,1\right)$ and $v\sim {\chi }^{2}\left(r\right)$ and are stochastically independent. Then the random variable $t=\frac{w}{\sqrt{\frac{v}{r}}}$ has the t-distribution with r degrees of freedom . The pdf and cumulative function of t are $f\left(t\right)=\frac{\Gamma \left(\frac{r+1}{2}\right)}{\sqrt{r\pi }\Gamma \left(\frac{r}{2}\right)}{\left(1+\frac{{t}^{2}}{r}\right)}^{-\left(\frac{r+1}{2}\right)}$ and $F\left(t\right)=\frac{1}{2}+t\Gamma \left(\frac{t}{2}\right).$ The mean and variance of the distribution are 0 for $r>1$ and $\frac{r}{r-2}$ for $t>2,$ respectively. The mean of the t-distribution is undefined for $t\le 1.$ The variance of the distribution is $\infty$ for $1 and undefined for $r\le 1.$ The t-distribution plays a prominent role in hypothesis testing that is well-known to all undergraduate economics majors.

## F distribution.

Consider two stochastically independent chi-square random variable such that $u\sim {\text{χ}}^{2}\left({r}_{1}\right)$ and $v\sim {\text{χ}}^{2}\left({r}_{2}\right)$ and $u,v>0.$ The new random variable $f=\frac{u}{{r}_{1}}}{v}{{r}_{2}}}$ has a F-distribution with ${r}_{1}$ and ${r}_{2}$ degrees of freedom. The pdf for the F-distribution is $g\left(f\right)=\frac{\Gamma \left(\frac{{r}_{1}+{r}_{2}}{2}\right)\left(\frac{{r}_{1}}{{r}_{2}}\right)}{\Gamma \left(\frac{{r}_{1}}{2}\right)\Gamma \left(\frac{{r}_{2}}{2}\right)}\frac{{f}^{\frac{{r}_{1}}{2}-1}}{{\left(1+\frac{{r}_{1}f}{{r}_{2}}\right)}^{\frac{{r}_{1}+{r}_{2}}{2}}}.$ The F-distribution is used in testing if population variances are equal and in performing likelihood ratio tests.

## Multinomial distribution.

Consider the n random variables ${x}_{1},{x}_{2},\cdots ,{x}_{n}$ where each variable has a normal distribution—that is, ${x}_{i}\sim N\left({\mu }_{i},{\sigma }_{i}^{2}\right)$ and the covariance between of the variables is ${\sigma }_{ij}=E\left[\left({x}_{i}-{\mu }_{i}\right)\left({x}_{j}-{\mu }_{j}.\right)\right]$ We can arrange the variances and covariances into a n -by- n matrix where $\Sigma =\left[\begin{array}{cccc}{\sigma }_{1}^{2}& {\sigma }_{12}& \cdots & {\sigma }_{1n}\\ {\sigma }_{21}& {\sigma }_{2}^{2}& \cdots & {\sigma }_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\sigma }_{n1}& {\sigma }_{n2}& \cdots & {\sigma }_{n}^{2}\end{array}\right]$ that is known as the variance-covariance matrix. Define the vector $\left(x-\mu \right)=\left(\begin{array}{c}{x}_{1}-{\mu }_{1}\\ ⋮\\ {x}_{n}-{\mu }_{n}\end{array}\right)$ and ${\left(x-\mu \right)}^{\prime }$ as its transpose. Then, ${\left(x-\mu \right)}^{\prime }\Sigma \left(x-\mu \right)=\sum _{i=1}^{n}\sum _{j=1}^{n}\left({x}_{i}-{\mu }_{i}\right)\left({x}_{j}-{\mu }_{j}\right){\sigma }_{ij},$ where ${\sigma }_{ii}={\sigma }_{i}^{2}.$ If $|\Sigma |$ is the determinant of the variance-covariance matrix, then the pdf for the joint distribution of these random variables is $f\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)=\frac{1}{{\left(2\pi \right)}^{n/2}{|\Sigma |}^{1}{2}}}{e}^{-\frac{1}{2}{\left(x-\mu \right)}^{\prime }\Sigma \left(x-\mu \right)}.$ If the random variables are stochastically independent the covariances are equal to 0 and the pdf becomes $f\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)=\frac{1}{{\left(2\pi \right)}^{n/2}{\left(\prod _{i=1}^{n}{\sigma }_{1}^{2}\right)}^{1}{2}}}{e}^{-\frac{1}{2}\sum _{i=1}^{n}\frac{{\left({x}_{i}-{\mu }_{i}\right)}^{2}}{{\sigma }_{i}^{2}}}.$ If the n random variables are all drawn from the same normal distribution with a mean of μ and a variance of ${\sigma }^{2},$ then the pdf simplifies to $f\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)=\frac{1}{{\left(2\pi {\sigma }^{2}\right)}^{n/2}}{e}^{-\frac{1}{2{\sigma }^{2}}\sum _{i=1}^{n}{\left({x}_{i}-\mu \right)}^{2}}.$

## Bias.

The bias of an estimator is defined to be $B\left(\stackrel{^}{\theta }\right)=E\left(\stackrel{^}{\theta }\right)-\theta .$ An estimator is unbiased if and only if $B\left(\stackrel{^}{\theta }\right)=0.$

## Mean square error.

The mean square error (MSE) of an estimator is defined to be $MSE\left(\stackrel{^}{\theta }\right)=E\left[{\left(\stackrel{^}{\theta }-\theta \right)}^{2}\right].$ It is relatively easy to show that $MSE\left(\stackrel{^}{\theta }\right)=V\left(\stackrel{^}{\theta }\right)+{\left(B\left(\stackrel{^}{\theta }\right)\right)}^{2}.$ Often a biased estimator with a smaller MSE may be preferred to an unbiased estimator with a relatively larger MSE.

## Efficiency.

An estimator $\stackrel{^}{\theta }$ is relatively more efficient than $\stackrel{˜}{\theta }$ if and only if $V\left(\stackrel{^}{\theta }\right) Generally, we would prefer to use the most efficient estimator available (if it is unbiased).

## Plim.

${x}_{n}$ converges to a constant, c , if ${\mathrm{lim}}_{n\to \infty }\mathrm{Pr}\left(|{x}_{n}-c|>\epsilon \right)=0$ for any positive $\epsilon .$ We can write this relationship as $p\mathrm{lim}{x}_{n}=c.$

Greene Greene, William H. (1990). Econometric Analysis (New York: Macmillan Publishing Company): 103. offers this example of plim: Suppose ${x}_{n}$ equals 0 with probability $1-\left(\frac{1}{n}\right)$ and n with probability $\left(\frac{1}{n}\right).$ As n increases, the second point becomes more remote from the first point. However, at the same time the probability of observing the second point becomes more and more unlikely. This effect is shown in Figure 5 where as n increases the probability distribution concentrates more and more on 1.

## Consistency.

The estimator $\stackrel{^}{\theta }$ is a consistent estimator of θ if and only if $p\mathrm{lim}\stackrel{^}{\theta }=\theta .$

## Asymmtotically unbiased.

An estimator $\stackrel{^}{\theta }$ is an asymtotically unbiased estimator of θ if ${\mathrm{lim}}_{n\to \infty }E\left[\stackrel{^}{\theta }\right]=\theta .$

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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