Home
Econometrics for honors students Background issues in statistics Statistical terminology
E
(
a
)
=
a
,
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadweadaqadaqaaiaadggaaiaawIcacaGLPaaacqGH9aqpcaWGHbGaaiilaaaa@3BBE@
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@
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
f
(
x
,
y
)
≥
0
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqadaqaaiaadIhacaGGSaGaamyEaaGaayjkaiaawMcaaiabgwMiZkaaicdaaaa@3D88@ for all
x and
y and
∫
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@
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.
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.
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.
