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  1. E ( a ) = a , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadweadaqadaqaaiaadggaaiaawIcacaGLPaaacqGH9aqpcaWGHbGaaiilaaaa@3BBE@
  2. E ( a x 2 + b x + c ) = a E ( x 2 ) + b μ + c . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadweadaqadaqaaiaadggacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyaiaadIhacqGHRaWkcaWGJbaacaGLOaGaayzkaaGaeyypa0JaamyyaiaadweadaqadaqaaiaadIhadaahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaacqGHRaWkcaWGIbGaeqiVd0Maey4kaSIaam4yaiaac6caaaa@4BCC@
  3. E ( a x + b ) = a μ + b , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadweadaqadaqaaiaadggacaWG4bGaey4kaSIaamOyaaGaayjkaiaawMcaaiabg2da9iaadggacqaH8oqBcqGHRaWkcaWGIbGaaiilaaaa@4203@

These rules work both for discrete and continuous random variables.

Joint distributions

The joint pdf for two random variables.

Any function, f ( x , y ) , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqadaqaaiaadIhacaGGSaGaamyEaaGaayjkaiaawMcaaiaacYcaaaa@3BB8@ that has the characteristics

  1. f ( x , y ) 0 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqadaqaaiaadIhacaGGSaGaamyEaaGaayjkaiaawMcaaiabgwMiZkaaicdaaaa@3D88@ for all x and y and
  2. y x f ( x , y ) d x d y = 1 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaapefabaWaa8quaeaacaWGMbWaaeWaaeaacaWG4bGaaiilaiaadMhaaiaawIcacaGLPaaacaWGKbGaamiEaaWcbaGaamiEaaqab0Gaey4kIipakiaadsgacaWG5baaleaacaWG5baabeqdcqGHRiI8aOGaeyypa0JaaGymaaaa@4739@

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, C o v ( x , y ) MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeacaWGVbGaamODamaabmaabaGaamiEaiaacYcacaWG5baacaGLOaGaayzkaaaaaa@3CD4@ or σ x y , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBaaaleaacaWG4bGaamyEaaqabaGccaGGSaaaaa@3A8D@ is defined to be C o v ( x , y ) = E [ ( x μ x ) ( y μ y ) ] . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeacaWGVbGaamODamaabmaabaGaamiEaiaacYcacaWG5baacaGLOaGaayzkaaGaeyypa0JaamyramaadmaabaWaaeWaaeaacaWG4bGaeyOeI0IaeqiVd02aaSbaaSqaaiaadIhaaeqaaaGccaGLOaGaayzkaaWaaeWaaeaacaWG5bGaeyOeI0IaeqiVd02aaSbaaSqaaiaadMhaaeqaaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaGaaiOlaaaa@4E02@ Expansion gives the alternative definition that σ x y = E ( x y ) μ x μ y . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBaaaleaacaWG4bGaamyEaaqabaGccqGH9aqpcaWGfbWaaeWaaeaacaWG4bGaamyEaaGaayjkaiaawMcaaiabgkHiTiabeY7aTnaaBaaaleaacaWG4baabeaakiabeY7aTnaaBaaaleaacaWG5baabeaakiaac6caaaa@46A3@

Stochastic independence.

The random variables x and y are stochastically independent if and only if σ x y = 0. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBaaaleaacaWG4bGaamyEaaqabaGccqGH9aqpcaaIWaGaaiOlaaaa@3C4F@ An equivalent definition of independence is that x and y are stochastically independent if and only if f ( x , y ) = f ( x ) f ( y ) , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqadaqaaiaadIhacaGGSaGaamyEaaGaayjkaiaawMcaaiabg2da9iaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacaWGMbWaaeWaaeaacaWG5baacaGLOaGaayzkaaGaaiilaaaa@43A1@ 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 ( x y ) = μ x μ y . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadweadaqadaqaaiaadIhacaWG5baacaGLOaGaayzkaaGaeyypa0JaeqiVd02aaSbaaSqaaiaadIhaaeqaaOGaeqiVd02aaSbaaSqaaiaadMhaaeqaaOGaaiOlaaaa@41C2@

Correlation coefficient.

The correlation coefficient, ρ , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg8aYjaacYcaaaa@3859@ is defined to be ρ x y = σ x y σ x σ y . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg8aYnaaBaaaleaacaWG4bGaamyEaaqabaGccqGH9aqpdaWcaaqaaiabeo8aZnaaBaaaleaacaWG4bGaamyEaaqabaaakeaacqaHdpWCdaWgaaWcbaGaamiEaaqabaGccqaHdpWCdaWgaaWcbaGaamyEaaqabaaaaOGaaiOlaaaa@4583@ The correlation coefficient is a unitless number that varies between -1 and +1. Clearly, two random variables are stochastically independent if and only if ρ x y = 0. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg8aYnaaBaaaleaacaWG4bGaamyEaaqabaGccqGH9aqpcaaIWaGaaiOlaaaa@3C4C@

Discrete distributions

Binomial distribution.

The discrete random variable x has a binomial distribution if f ( x ) = { ( n x ) p x ( 1 p ) n x ,    x = 0 , 1 , , n 0   elsewhere MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpdaGabaabaeqabaWaaeWaaqaabeqaaiaad6gaaeaacaWG4baaaiaawIcacaGLPaaacaWGWbWaaWbaaSqabeaacaWG4baaaOWaaeWaaeaacaaIXaGaeyOeI0IaamiCaaGaayjkaiaawMcaamaaCaaaleqabaGaamOBaiabgkHiTiaadIhaaaGccaGGSaGaaeiiaiaabccacaWG4bGaeyypa0JaaGimaiaacYcacaaIXaGaaiilaiablAciljaacYcacaWGUbaabaGaaGimaiaabccacaqGGaGaaeyzaiaabYgacaqGZbGaaeyzaiaabEhacaqGObGaaeyzaiaabkhacaqGLbaaaiaawUhaaaaa@5C55@ where ( n x ) = n ! x ! ( n x ) . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaaeaqabeaacaWGUbaabaGaamiEaaaacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacaWGUbGaaiyiaaqaaiaadIhacaGGHaWaaeWaaeaacaWGUbGaeyOeI0IaamiEaaGaayjkaiaawMcaaaaacaGGUaaaaa@42D1@ For the binomial distribution, μ = n p MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTjabg2da9iaad6gacaWGWbaaaa@3A8D@ and σ 2 = n p ( 1 p ) . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaCaaaleqabaGaaGOmaaaakiabg2da9iaad6gacaWGWbWaaeWaaeaacaaIXaGaeyOeI0IaamiCaaGaayjkaiaawMcaaiaac6caaaa@4065@

Uniform distribution.

The discrete random variable x has a uniform distribution if f ( x ) = { 1 b a  if  a x b 0   elsewhere } . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpdaGadaabaeqabaWaaSaaaeaacaaIXaaabaGaamOyaiabgkHiTiaadggaaaGaaeiiaiaabMgacaqGMbGaaeiiaiaadggacqGHKjYOcaWG4bGaeyizImQaamOyaaqaaiaaicdacaqGGaGaaeiiaiaabwgacaqGSbGaae4CaiaabwgacaqG3bGaaeiAaiaabwgacaqGYbGaaeyzaaaacaGL7bGaayzFaaGaaiOlaaaa@547D@ The mean and variance of the uniform distribution are μ = a + b 2 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTjabg2da9maalaaabaGaamyyaiabgUcaRiaadkgaaeaacaaIYaaaaaaa@3C20@ and σ 2 = ( b a ) 2 12 . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaCaaaleqabaGaaGOmaaaakiabg2da9maalaaabaWaaeWaaeaacaWGIbGaeyOeI0IaamyyaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOqaaiaaigdacaaIYaaaaiaac6caaaa@4114@

Poisson distribution.

The discrete random variable x has a Poisson distribution if f ( x ) = { m x e m x ! ,    x = 0 , 1 , 0   elsewhere MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpdaGabaabaeqabaWaaSaaaeaacaWGTbWaaWbaaSqabeaacaWG4baaaOGaamyzamaaCaaaleqabaGaeyOeI0IaamyBaaaaaOqaaiaadIhacaGGHaaaaiaacYcacaqGGaGaaeiiaiaadIhacqGH9aqpcaaIWaGaaiilaiaaigdacaGGSaGaeSOjGSeabaGaaGimaiaabccacaqGGaGaaeyzaiaabYgacaqGZbGaaeyzaiaabEhacaqGObGaaeyzaiaabkhacaqGLbaaaiaawUhaaaaa@54A7@ For the Poisson distribution μ = σ 2 = m . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTjabg2da9iabeo8aZnaaCaaaleqabaGaaGOmaaaakiabg2da9iaad2gacaGGUaaaaa@3E05@ 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 ( x ) = { λ e λ x ,  for  x 0    0   for  x < 0 } . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpdaGadaabaeqabaGaeq4UdWMaamyzamaaCaaaleqabaGaeyOeI0Iaeq4UdWMaamiEaaaakiaacYcacaqGGaGaaeOzaiaab+gacaqGYbGaaeiiaiaadIhacqGHLjYScaaIWaGaaeiiaiaabccaaeaacaqGWaGaaeiiaiaabccacaqGGaGaaeOzaiaab+gacaqGYbGaaeiiaiaadIhacqGH8aapcaaIWaaaaiaawUhacaGL9baacaGGUaaaaa@560F@ The cumulative exponential distribution is given by F ( x ) = 1 e λ x , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpcaaIXaGaeyOeI0IaamyzamaaCaaaleqabaGaeyOeI0Iaeq4UdWMaamiEaaaakiaacYcaaaa@4157@ for x 0. MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacqGHLjYScaaIWaGaaiOlaaaa@3A18@ 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 μ = λ 1 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTjabg2da9iabeU7aSnaaCaaaleqabaGaeyOeI0IaaGymaaaaaaa@3C2E@ and σ 2 = λ 2 . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaCaaaleqabaGaaGOmaaaakiabg2da9iabeU7aSnaaCaaaleqabaGaeyOeI0IaaGOmaaaakiaac6caaaa@3DEB@

Cauchy distribution.

A random variable x , where < x < , MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiabg6HiLkabgYda8iaadIhacqGH8aapcqGHEisPcaGGSaaaaa@3D6D@ has a Cauchy (or Cauchy-Lorentz) distribution if its pdf is f ( x ) = 1 π [ γ ( x x 0 ) 2 + γ 2 ] . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaaigdaaeaacqaHapaCaaWaamWaaeaadaWcaaqaaiabeo7aNbqaamaabmaabaGaamiEaiabgkHiTiaadIhadaWgaaWcbaGaaGimaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcqaHZoWzdaahaaWcbeqaaiaaikdaaaaaaaGccaGLBbGaayzxaaGaaiOlaaaa@4B12@ The parameter x 0 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaaGimaaqabaaaaa@37CC@ 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.

The cauchy distribution.

Graph of the Cauchy distribution for two values of the parameters.
The two panels represent the Cauchy distribution for two sets of values of x 0 and γ .

Normal distribution.

The continuous random variable x has a normal distribution with a mean of μ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTbaa@379F@ and a variance of σ 2 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaCaaaleqabaGaaGOmaaaaaaa@3895@ if its pdf is f ( x ) = 1 σ 2 π e ( x μ ) 2 2 σ 2 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaaigdaaeaacqaHdpWCdaGcaaqaaiaaikdacqaHapaCaSqabaaaaOGaamyzamaaCaaaleqabaGaeyOeI0YaaSaaaeaadaqadaqaaiaadIhacqGHsislcqaH8oqBaiaawIcacaGLPaaadaahaaadbeqaaiaaikdaaaaaleaacaaIYaGaeq4Wdm3aaWbaaWqabeaacaaIYaaaaaaaaaaaaa@4B27@ for x . MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiabg6HiLkabgsMiJkaadIhacqGHKjYOcqGHEisPcaGGUaaaaa@3ED1@ The distribution is symmetric around the mean.

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
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Source:  OpenStax, Econometrics for honors students. OpenStax CNX. Jul 20, 2010 Download for free at http://cnx.org/content/col11208/1.2
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