The continuous random variable x has an exponential distribution if
f(
x
)={
λ
e
−λx
, for x≥0
0 for x<0
}.
f(
x
)={
λ
e
−λx
, 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
−λx
,
F(
x
)=1−
e
−λx
,
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpcaaIXaGaeyOeI0IaamyzamaaCaaaleqabaGaeyOeI0Iaeq4UdWMaamiEaaaakiaacYcaaaa@4157@
for
x≥0.
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
μ=
λ
−1
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTjabg2da9iabeU7aSnaaCaaaleqabaGaeyOeI0IaaGymaaaaaaa@3C2E@
and
σ
2
=
λ
−2
.
σ
2
=
λ
−2
.
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaCaaaleqabaGaaGOmaaaakiabg2da9iabeU7aSnaaCaaaleqabaGaeyOeI0IaaGOmaaaakiaac6caaaa@3DEB@
A random variable x, where
−∞<x<∞,
−∞<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
].
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
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 continuous random variable x has a normal distribution with a mean of
μ
μ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTbaa@379F@
and a variance of
σ
2
σ
2
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaCaaaleqabaGaaGOmaaaaaaa@3895@
if its pdf is
f(
x
)=
1
σ
2π
e
−
(
x−μ
)
2
2
σ
2
f(
x
)=
1
σ
2π
e
−
(
x−μ
)
2
2
σ
2
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaaigdaaeaacqaHdpWCdaGcaaqaaiaaikdacqaHapaCaSqabaaaaOGaamyzamaaCaaaleqabaGaeyOeI0YaaSaaaeaadaqadaqaaiaadIhacqGHsislcqaH8oqBaiaawIcacaGLPaaadaahaaadbeqaaiaaikdaaaaaleaacaaIYaGaeq4Wdm3aaWbaaWqabeaacaaIYaaaaaaaaaaaaa@4B27@
for
−∞≤x≤∞.
−∞≤x≤∞.
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiabg6HiLkabgsMiJkaadIhacqGHKjYOcqGHEisPcaGGUaaaaa@3ED1@
The distribution is symmetric around the mean.
The continuous random variable x has log normal distribution if y has a normal distribution and
x=
e
y
.
x=
e
y
.
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacqGH9aqpcaWGLbWaaWbaaSqabeaacaWG5baaaOGaaiOlaaaa@3ABD@
Thus, if
y∼N(
μ,
σ
2
),
y∼N(
μ,
σ
2
),
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhacqWI8iIocaWGobWaaeWaaeaacqaH8oqBcaGGSaGaeq4Wdm3aaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaGaaiilaaaa@4038@
then the pdf of a log normal distribution is
f(
x
)={
1
xσ
2π
e
−
(
ln(
x
)−μ
)
2
2
σ
2
, for x>0
0 otherwise
}.
f(
x
)={
1
xσ
2π
e
−
(
ln(
x
)−μ
)
2
2
σ
2
, for x>0
0 otherwise
}.
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@6667@
The mean and variance of x are
μ
x
=
e
μ+
σ
2
2
μ
x
=
e
μ+
σ
2
2
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTnaaBaaaleaacaWG4baabeaakiabg2da9iaadwgadaahaaWcbeqaaiabeY7aTjabgUcaRmaaleaameaacqaHdpWCdaahaaqabeaacaaIYaaaaaqaaiaaikdaaaaaaaaa@4101@
and
σ
x
2
=(
e
σ
2
−1
)
e
2μ+
σ
2
.
σ
x
2
=(
e
σ
2
−1
)
e
2μ+
σ
2
.
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaDaaaleaacaWG4baabaGaaGOmaaaakiabg2da9maabmaabaGaamyzamaaCaaaleqabaGaeq4Wdm3aaWbaaWqabeaacaaIYaaaaaaakiabgkHiTiaaigdaaiaawIcacaGLPaaacaWGLbWaaWbaaSqabeaacaaIYaGaeqiVd0Maey4kaSIaeq4Wdm3aaWbaaWqabeaacaaIYaaaaaaakiaac6caaaa@4975@
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.
A positive random variable x has a gamma distribution if its pdf is
f(
x
)=
1
Γ(
α
)
β
α
x
α−1
e
−
x
β
f(
x
)=
1
Γ(
α
)
β
α
x
α−1
e
−
x
β
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaaigdaaeaacqqHtoWrdaqadaqaaiabeg7aHbGaayjkaiaawMcaaiabek7aInaaCaaaleqabaGaeqySdegaaaaakiaadIhadaahaaWcbeqaaiabeg7aHjabgkHiTiaaigdaaaGccaWGLbWaaWbaaSqabeaacqGHsisldaWcbaadbaGaamiEaaqaaiabek7aIbaaaaaaaa@4C6C@
for
x>0
x>0
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacqGH+aGpcaaIWaaaaa@38A8@
and 0 elsewhere.
Γ(
α
)
Γ(
α
)
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfo5ahnaabmaabaGaeqySdegacaGLOaGaayzkaaaaaa@3A79@
is known as the gamma function and is defined to be
Γ(
α
)=
∫
0
∞
y
α−1
e
−y
dy
=(
α−1
)!.
Γ(
α
)=
∫
0
∞
y
α−1
e
−y
dy
=(
α−1
)!.
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfo5ahnaabmaabaGaeqySdegacaGLOaGaayzkaaGaeyypa0Zaa8qmaeaacaWG5bWaaWbaaSqabeaacqaHXoqycqGHsislcaaIXaaaaOGaamyzamaaCaaaleqabaGaeyOeI0IaamyEaaaakiaadsgacaWG5baaleaacaaIWaaabaGaeyOhIukaniabgUIiYdGccqGH9aqpdaqadaqaaiabeg7aHjabgkHiTiaaigdaaiaawIcacaGLPaaacaGGHaGaaiOlaaaa@5079@
The gamma function is often used to model waiting times like waiting for death. Its mean and variance are given by
μ=αβ
μ=αβ
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTjabg2da9iabeg7aHjabek7aIbaa@3BE5@
and
σ
2
=α
β
2
.
σ
2
=α
β
2
.
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaCaaaleqabaGaaGOmaaaakiabg2da9iabeg7aHjabek7aInaaCaaaleqabaGaaGOmaaaakiaac6caaaa@3E8A@
A chi-square distribution ( χ 2 ( k) χ 2 ( k)MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeE8aJnaaCaaaleqabaGaaGOmaaaakmaabmaabaGaam4AaaGaayjkaiaawMcaaiaac6caaaa@3BBE@ ) is the sum of k independent standard normal random variables and is a special case of the gamma distribution (with
α=
k
2
α=
k
2
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHjabg2da9maalaaabaGaam4Aaaqaaiaaikdaaaaaaa@3A4A@
and
β=2
β=2
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabek7aIjabg2da9iaaikdaaaa@394C@
). The pdf of a chi-square distribution with k degrees of freedom is
f(
x
)=
1
2
k
2
Γ(
k
2
)
x
k
2
−1
e
−
x
2
f(
x
)=
1
2
k
2
Γ(
k
2
)
x
k
2
−1
e
−
x
2
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaWaaWbaaSqabeaadaWcdaadbaGaam4AaaqaaiaaikdaaaaaaOGaeu4KdC0aaeWaaeaadaWcbaWcbaGaam4AaaqaaiaaikdaaaaakiaawIcacaGLPaaaaaGaamiEamaaCaaaleqabaWaaSqaaWqaaiaadUgaaeaacaaIYaaaaSGaeyOeI0IaaGymaaaakiaadwgadaahaaWcbeqaaiabgkHiTmaaleaameaacaWG4baabaGaaGOmaaaaaaaaaa@4B36@
where x > 0. Its mean and variance are
μ=k
μ=k
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTjabg2da9iaadUgaaaa@3995@
and
σ
2
=2k.
σ
2
=2k.
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaCaaaleqabaGaaGOmaaaakiabg2da9iaaikdacaWGRbGaaiOlaaaa@3C03@
If
y=
∑
i=1
k
x
i
2
y=
∑
i=1
k
x
i
2
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhacqGH9aqpdaaeWbqaaiaadIhadaqhaaWcbaGaamyAaaqaaiaaikdaaaaabaGaamyAaiabg2da9iaaigdaaeaacaWGRbaaniabggHiLdaaaa@4097@
where the xi's are independently drawn from the standard normal distribution (N(1, 0)), then
y
i
∼
χ
2
(
k
).
y
i
∼
χ
2
(
k
).
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMhadaWgaaWcbaGaamyAaaqabaGccqWI8iIocqaHhpWydaahaaWcbeqaaiaaikdaaaGcdaqadaqaaiaadUgaaiaawIcacaGLPaaacaGGUaaaaa@3F09@
Consider two random variables, x and v. Assume that
x∼N(
0,1
)
x∼N(
0,1
)
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacqWI8iIocaWGobWaaeWaaeaacaaIWaGaaiilaiaaigdaaiaawIcacaGLPaaaaaa@3C90@
and
v∼
χ
2
(
r
)
v∼
χ
2
(
r
)
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAhacqWI8iIocqaHhpWydaahaaWcbeqaaiaaikdaaaGcdaqadaqaaiaadkhaaiaawIcacaGLPaaaaaa@3D37@
and are stochastically independent. Then the random variable
t=
w
v
r
t=
w
v
r
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshacqGH9aqpdaWcaaqaaiaadEhaaeaadaGcaaqaamaalaaabaGaamODaaqaaiaadkhaaaaaleqaaaaaaaa@3B11@
has the t-distribution with rdegrees of freedom. The pdf and cumulative function of t are
f(
t
)=
Γ(
r+1
2
)
rπ
Γ(
r
2
)
(
1+
t
2
r
)
−(
r+1
2
)
f(
t
)=
Γ(
r+1
2
)
rπ
Γ(
r
2
)
(
1+
t
2
r
)
−(
r+1
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqadaqaaiaadshaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiabfo5ahnaabmaabaWaaSaaaeaacaWGYbGaey4kaSIaaGymaaqaaiaaikdaaaaacaGLOaGaayzkaaaabaWaaOaaaeaacaWGYbGaeqiWdahaleqaaOGaeu4KdC0aaeWaaeaadaWcaaqaaiaadkhaaeaacaaIYaaaaaGaayjkaiaawMcaaaaadaqadaqaaiaaigdacqGHRaWkdaWcaaqaaiaadshadaahaaWcbeqaaiaaikdaaaaakeaacaWGYbaaaaGaayjkaiaawMcaamaaCaaaleqabaGaeyOeI0YaaeWaaeaadaWcaaqaaiaadkhacqGHRaWkcaaIXaaabaGaaGOmaaaaaiaawIcacaGLPaaaaaaaaa@5466@
and
F(
t
)=
1
2
+tΓ(
t
2
).
F(
t
)=
1
2
+tΓ(
t
2
).
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaqadaqaaiaadshaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaiabgUcaRiaadshacqqHtoWrdaqadaqaamaalaaabaGaamiDaaqaaiaaikdaaaaacaGLOaGaayzkaaGaaiOlaaaa@4306@
The mean and variance of the distribution are 0 for
r>1
r>1
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkhacqGH+aGpcaaIXaaaaa@38A3@
and
r
r−2
r
r−2
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaamOCaaqaaiaadkhacqGHsislcaaIYaaaaaaa@3990@
for
t>2,
t>2,
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadshacqGH+aGpcaaIYaGaaiilaaaa@3956@
respectively. The t-distribution plays a prominent role in hypothesis testing that is well-known to all undergraduate economics majors.
Consider two stochastically independent chi-square random variable such that
u∼
χ
2
(
r
1
)
u∼
χ
2
(
r
1
)
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhacqWI8iIocaqGhpWaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaWGYbWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaaaaa@3DBD@
and
v∼
χ
2
(
r
2
)
v∼
χ
2
(
r
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAhacqWI8iIocaqGhpWaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaWGYbWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaaaa@3DBF@
and
u,v>0.
u,v>0.
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhacaGGSaGaamODaiabg6da+iaaicdacaGGUaaaaa@3B02@
The new random variable
f=
u
r
1
v
r
2
f=
u
r
1
v
r
2
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgacqGH9aqpdaWcaaqaamaaliaabaGaamyDaaqaaiaadkhadaWgaaWcbaGaaGymaaqabaaaaaGcbaWaaSGaaeaacaWG2baabaGaamOCamaaBaaaleaacaaIYaaabeaaaaaaaaaa@3DCA@
has a F-distribution with
r
1
r
1
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkhadaWgaaWcbaGaaGymaaqabaaaaa@37C7@
and
r
2
r
2
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkhadaWgaaWcbaGaaGOmaaqabaaaaa@37C8@
degrees of freedom. The pdf for the F-distribution is
g(
f
)=
Γ(
r
1
+
r
2
2
)(
r
1
r
2
)
Γ(
r
1
2
)Γ(
r
2
2
)
f
r
1
2
−1
(
1+
r
1
f
r
2
)
r
1
+
r
2
2
.
g(
f
)=
Γ(
r
1
+
r
2
2
)(
r
1
r
2
)
Γ(
r
1
2
)Γ(
r
2
2
)
f
r
1
2
−1
(
1+
r
1
f
r
2
)
r
1
+
r
2
2
.
MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@6771@
The F-distribution is used in testing if population variances are equal and in performing likelihood ratio tests.
Consider the n random variables
x
1
,
x
2
,⋯,
x
n
x
1
,
x
2
,⋯,
x
n
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIYaaabeaakiaacYcacqWIVlctcaGGSaGaamiEamaaBaaaleaacaWGUbaabeaaaaa@3FE0@
where each variable has a normal distribution—that is,
x
i
∼N(
μ
i
,
σ
i
2
)
x
i
∼N(
μ
i
,
σ
i
2
)
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaamyAaaqabaGccqWI8iIocaWGobWaaeWaaeaacqaH8oqBdaWgaaWcbaGaamyAaaqabaGccaGGSaGaeq4Wdm3aa0baaSqaaiaadMgaaeaacaaIYaaaaaGccaGLOaGaayzkaaaaaa@42BD@
and the covariance between of the variables is
σ
ij
=E[
(
x
i
−
μ
i
)(
x
j
−
μ
j
.
)
]
σ
ij
=E[
(
x
i
−
μ
i
)(
x
j
−
μ
j
.
)
]
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpcaWGfbWaamWaaeaadaqadaqaaiaadIhadaWgaaWcbaGaamyAaaqabaGccqGHsislcqaH8oqBdaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaadaqadaqaaiaadIhadaWgaaWcbaGaamOAaaqabaGccqGHsislcqaH8oqBdaWgaaWcbaGaamOAaaqabaaakiaawIcacaGLPaaaaiaawUfacaGLDbaaaaa@4C65@
We can arrange the variances and covariances into a n-by-n matrix where
Σ=[
σ
1
2
σ
12
⋯
σ
1n
σ
21
σ
2
2
⋯
σ
2n
⋮
⋮
⋱
⋮
σ
n1
σ
n2
⋯
σ
n
2
]
Σ=[
σ
1
2
σ
12
⋯
σ
1n
σ
21
σ
2
2
⋯
σ
2n
⋮
⋮
⋱
⋮
σ
n1
σ
n2
⋯
σ
n
2
]
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@6816@
that is known as the variance-covariance matrix. Define the vector
(
x−μ
)=(
x
1
−
μ
1
⋮
x
n
−
μ
n
)
(
x−μ
)=(
x
1
−
μ
1
⋮
x
n
−
μ
n
)
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaaCiEaiabgkHiTiaahY7aaiaawIcacaGLPaaacqGH9aqpdaqadaqaauaabeqadeaaaeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaeqiVd02aaSbaaSqaaiaaigdaaeqaaaGcbaGaeSO7I0eabaGaamiEamaaBaaaleaacaWGUbaabeaakiabgkHiTiabeY7aTnaaBaaaleaacaWGUbaabeaaaaaakiaawIcacaGLPaaaaaa@4AA8@
and
(
x−μ
)
′
(
x−μ
)
′
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaaCiEaiabgkHiTiaahY7aaiaawIcacaGLPaaadaahaaWcbeqaaOGamai4gkdiIcaaaaa@3DC7@
as its transpose. Then,
(
x−μ
)
′
Σ(
x−μ
)=
∑
i=1
n
∑
j=1
n
(
x
i
−
μ
i
)(
x
j
−
μ
j
)
σ
ij
,
(
x−μ
)
′
Σ(
x−μ
)=
∑
i=1
n
∑
j=1
n
(
x
i
−
μ
i
)(
x
j
−
μ
j
)
σ
ij
,
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@63E3@
where
σ
ii
=
σ
i
2
.
σ
ii
=
σ
i
2
.
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamyAaaqabaGccqGH9aqpcqaHdpWCdaqhaaWcbaGaamyAaaqaaiaaikdaaaaaaa@3E5E@
If
| Σ |
| Σ |
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaaC4OdaGaay5bSlaawIa7aaaa@3A3A@
is the determinant of the variance-covariance matrix, then the pdf for the joint distribution of these random variables is
f(
x
1
,
x
2
,…,
x
n
)=
1
(
2π
)
n/2
| Σ |
1
2
e
−
1
2
(
x−μ
)
′
Σ(
x−μ
)
.
f(
x
1
,
x
2
,…,
x
n
)=
1
(
2π
)
n/2
| Σ |
1
2
e
−
1
2
(
x−μ
)
′
Σ(
x−μ
)
.
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaqadaqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIYaaabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaaiabg2da9maalaaabaGaaGymaaqaamaabmaabaGaaGOmaiabec8aWbGaayjkaiaawMcaamaaCaaaleqabaGaamOBaiaac+cacaaIYaaaaOWaaqWaaeaacaWHJoaacaGLhWUaayjcSdWaaWbaaSqabeaadaWccaqaaiaaigdaaeaacaaIYaaaaaaaaaGccaWGLbWaaWbaaSqabeaacqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaaamaabmaabaGaaCiEaiabgkHiTiaahY7aaiaawIcacaGLPaaadaahaaadbeqaaSGamaiUgkdiIcaacaWHJoWaaeWaaeaacaWH4bGaeyOeI0IaaCiVdaGaayjkaiaawMcaaaaakiaac6caaaa@61FC@
If the random variables are stochastically independent the covariances are equal to 0 and the pdf becomes
f(
x
1
,
x
2
,…,
x
n
)=
1
(
2π
)
n/2
(
∏
i=1
n
σ
1
2
)
1
2
e
−
1
2
∑
i=1
n
(
x
i
−
μ
i
)
2
σ
i
2
.
f(
x
1
,
x
2
,…,
x
n
)=
1
(
2π
)
n/2
(
∏
i=1
n
σ
1
2
)
1
2
e
−
1
2
∑
i=1
n
(
x
i
−
μ
i
)
2
σ
i
2
.
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@6CB6@
If the n random variables are all drawn from the same normal distribution with a mean of μ and a variance of
σ
2
,
σ
2
,
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaCaaaleqabaGaaGOmaaaakiaacYcaaaa@394F@
then the pdf simplifies to
f(
x
1
,
x
2
,…,
x
n
)=
1
(
2π
σ
2
)
n/2
e
−
1
2
σ
2
∑
i=1
n
(
x
i
−μ
)
2
.
f(
x
1
,
x
2
,…,
x
n
)=
1
(
2π
σ
2
)
n/2
e
−
1
2
σ
2
∑
i=1
n
(
x
i
−μ
)
2
.
MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@60C6@
