<< Chapter < Page Chapter >> Page >
This module offers a brief and generally intuitive introduction to maximum likelihood estimation methods. It is intended as a guide for advanced undergraduates.

The maximum likelihood method

Introduction

The maximum likelihood (ML) method is an alternative to ordinary least squares (OLS) and offers a more general approach to the problem of finding estimators of unknown population parameters. In these notes we present an intuitive introduction to the ML technique. We begin our discussion with a description of continuous random variables.

Continuous random variables

Assume that x MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F1@ is a continuous random variable over the interval x . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaeyOhIuQaeyizImQaamiEaiabgsMiJkabg6HiLkaac6caaaa@3EDC@ Because of the assumption of continuity we need some special definitions.

Probability density function . Any function f ( x ) MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaaaa@3965@ that has the following characteristics is a probability density function (pdf): (1) f ( x ) > 0 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiabg6da+iaaicdaaaa@3B27@ and (2) f ( x ) d x = 1. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacaWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaamizaiaadIhaaSqaaiabgkHiTiabg6HiLcqaaiabg6HiLcqdcqGHRiI8aOGaeyypa0JaaGymaiaac6caaaa@4400@ The probability that x has a value between a and b is given by Pr ( a x b ) = a b f ( x ) d x . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiuaiaackhadaqadaqaaiaadggacqGHKjYOcaWG4bGaeyizImQaamOyaaGaayjkaiaawMcaaiabg2da9maapehabaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiaadsgacaWG4baaleaacaWGHbaabaGaamOyaaqdcqGHRiI8aOGaaiOlaaaa@4ACB@ Here are two examples of the probability density functions (pdf) of continuous random variables.

Uniform distribution

Let f ( x ) = 1 α MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGymaaqaaiabeg7aHbaaaaa@3CD5@ for 0 x α MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgsMiJkaadIhacqGHKjYOcqaHXoqyaaa@3CB4@ and 0 elsewhere, where α > 0. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeyOpa4JaaGimaiaac6caaaa@3A07@ A graph of the pdf for this distribution is shown in Figure 1.

Probability distribution function of a uniform distribution.

A graph of the uniform distribution.
The probability x falls between a and b is given by the colored in area.

It is easy to see from the graph that f ( x ) = 1 α > 0 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGymaaqaaiabeg7aHbaacqGH+aGpcaaIWaaaaa@3E97@ and Pr ( a x b ) = f ( x ) d x = 0 α 1 α d x = 1. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiiaiaabccaciGGqbGaaiOCamaabmaabaGaamyyaiabgsMiJkaadIhacqGHKjYOcaWGIbaacaGLOaGaayzkaaGaeyypa0Zaa8qCaeaacaWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaamizaiaadIhaaSqaaiabgkHiTiabg6HiLcqaaiabg6HiLcqdcqGHRiI8aOGaeyypa0Zaa8qCaeaadaWcaaqaaiaaigdaaeaacqaHXoqyaaGaamizaiaadIhaaSqaaiaaicdaaeaacqaHXoqya0Gaey4kIipakiabg2da9iaaigdacaGGUaaaaa@59F7@ Moreover, as shown in Figure 1, the area under the pdf curve between a and b is equal to the probability that x lies between a and b ; that is, Pr ( a x b ) = a b ( 1 α ) d x = x α | a b = b a α . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiuaiaackhadaqadaqaaiaadggacqGHKjYOcaWG4bGaeyizImQaamOyaaGaayjkaiaawMcaaiabg2da9maapehabaWaaeWaaeaadaWcaaqaaiaaigdaaeaacqaHXoqyaaaacaGLOaGaayzkaaGaamizaiaadIhaaSqaaiaadggaaeaacaWGIbaaniabgUIiYdGccqGH9aqpdaabcaqaamaalaaabaGaamiEaaqaaiabeg7aHbaaaiaawIa7amaaDaaaleaacaWGHbaabaGaamOyaaaakiabg2da9maalaaabaGaamOyaiabgkHiTiaadggaaeaacqaHXoqyaaGaaiOlaaaa@5809@

The calculation of the mean and variance of this distribution is relatively simple. The population mean is given by μ x = E ( x ) = 0 α x f ( x ) d x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaSbaaSqaaiaadIhaaeqaaOGaeyypa0JaamyramaabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maapehabaGaamiEaiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacaWGKbGaamiEaaWcbaGaaGimaaqaaiabeg7aHbqdcqGHRiI8aaaa@494E@ or μ x = 0 α x ( 1 α ) d x = x 2 2 α | 0 α = α 2 . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaSbaaSqaaiaadIhaaeqaaOGaeyypa0Zaa8qCaeaacaWG4bWaaeWaaeaadaWcaaqaaiaaigdaaeaacqaHXoqyaaaacaGLOaGaayzkaaGaamizaiaadIhaaSqaaiaaicdaaeaacqaHXoqya0Gaey4kIipakiabg2da9maaeiaabaWaaSaaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaiabeg7aHbaaaiaawIa7amaaDaaaleaacaaIWaaabaGaeqySdegaaOGaeyypa0ZaaSaaaeaacqaHXoqyaeaacaaIYaaaaaaa@527C@

The population variance Quite often, as in the exercises at the end of this module, it is easier to calculate the variance of a distribution using the alternative formula for the variance: σ x 2 = V ( x ) = E ( x μ ) 2 = E ( x 2 ) μ 2 , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaDaaaleaacaWG4baabaGaaGOmaaaakiabg2da9iaadAfadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpcaWGfbWaaeWaaeaacaWG4bGaeyOeI0IaeqiVd0gacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaamyramaabmaabaGaamiEamaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaiabgkHiTiabeY7aTnaaCaaaleqabaGaaGOmaaaakiaacYcaaaa@4F7E@ where E ( x 2 ) = x 2 f ( x ) d x . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadweadaqadaqaaiaadIhadaahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaacqGH9aqpdaWdbaqaaiaadIhadaahaaWcbeqaaiaaikdaaaGccaWGMbWaaeWaaeaacaWG4baacaGLOaGaayzkaaGaamizaiaadIhaaSqabeqaniabgUIiYdGccaGGUaaaaa@4530@ is given by V ( x ) = E [ ( x μ x ) 2 . ] MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaadweadaWadaqaamaabmaabaGaamiEaiabgkHiTiabeY7aTnaaBaaaleaacaWG4baabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOGaay5waiaaw2faaaaa@4465@ Thus, V ( x ) = 0 α ( x α 2 ) 2 ( 1 α ) d x = 0 α ( x 2 α x + α 2 4 ) ( 1 α ) d x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@62B5@ or V ( x ) = x 3 3 α x 2 2 + α 4 x | 0 α = α 2 3 α 2 2 + α 2 4 = α 2 12 . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5D95@

Because of the simple mathematical form of the uniform pdf, the calculations in Example 1 are relatively straight forward. While the calculations for random variables with a pdf that has a more complicated form are generally more difficult (if algebraically possible), the basic methodology remains the same. Example 2 considers the case of a more complicated pdf.

The normal distribution.

A random variable with a mean of μ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0gaaa@37AA@ and a variance of σ 2 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaWbaaSqabeaacaaIYaaaaaaa@38A0@ that has a normal distribution —that is, x ~ N ( μ , σ 2 ) MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaac6hacaWGobWaaeWaaeaacqaH8oqBcaGGSaGaeq4Wdm3aaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaGaaiifGaaa@4023@ has the pdf f ( x ) = 1 σ 2 π e ( x μ ) 2 2 σ 2 . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGymaaqaaiabeo8aZnaakaaabaGaaGOmaiabec8aWbWcbeaaaaGccaWGLbWaaWbaaSqabeaacqGHsisldaWcaaqaamaabmaabaGaamiEaiabgkHiTiabeY7aTbGaayjkaiaawMcaamaaCaaameqabaGaaGOmaaaaaSqaaiaaikdacqaHdpWCdaahaaadbeqaaiaaikdaaaaaaaaakiaac6caaaa@4BED@ A typical graph of this pdf is given in Figure 2. The area under the curve between values of x of a and b is equal to the probability that x falls between a and b .

Probability distribution function of a normal distribution.

A graph of the Normal distribution.
The probability x falls between a and b is given by the shaded area.

Joint distributions of samples and the ml method.

Most of the statistical work that economists use involves the use of a sample of observations. It is usual to assume that the members of the sample are drawn independently of each other. The implication of this assumption is that the pdf of the joint distribution is equal to the product of the pfd of each observation ; i.e.,

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Econometrics for honors students. OpenStax CNX. Jul 20, 2010 Download for free at http://cnx.org/content/col11208/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Econometrics for honors students' conversation and receive update notifications?

Ask