Continuous probability distribution
Fisher–Snedecor
Probability density function
Cumulative distribution function
Parameters
d 1 , d 2 > 0 deg. of freedom Support
x
∈
(
0
,
+
∞
)
{\displaystyle x\in (0,+\infty )\;}
if
d
1
=
1
{\displaystyle d_{1}=1}
, otherwise
x
∈
[
0
,
+
∞
)
{\displaystyle x\in [0,+\infty )\;}
PDF
(
d
1
x
)
d
1
d
2
d
2
(
d
1
x
+
d
2
)
d
1
+
d
2
x
B
(
d
1
2
,
d
2
2
)
{\displaystyle {\frac {\sqrt {\frac {(d_{1}x)^{d_{1}}d_{2}^{d_{2}}}{(d_{1}x+d_{2})^{d_{1}+d_{2}}}}}{x\,\mathrm {B} \!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\!}
CDF
I
d
1
x
d
1
x
+
d
2
(
d
1
2
,
d
2
2
)
{\displaystyle I_{\frac {d_{1}x}{d_{1}x+d_{2}}}\left({\tfrac {d_{1}}{2}},{\tfrac {d_{2}}{2}}\right)}
Mean
d
2
d
2
−
2
{\displaystyle {\frac {d_{2}}{d_{2}-2}}\!}
for d 2 > 2 Mode
d
1
−
2
d
1
d
2
d
2
+
2
{\displaystyle {\frac {d_{1}-2}{d_{1}}}\;{\frac {d_{2}}{d_{2}+2}}}
for d 1 > 2 Variance
2
d
2
2
(
d
1
+
d
2
−
2
)
d
1
(
d
2
−
2
)
2
(
d
2
−
4
)
{\displaystyle {\frac {2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)}}\!}
for d 2 > 4 Skewness
(
2
d
1
+
d
2
−
2
)
8
(
d
2
−
4
)
(
d
2
−
6
)
d
1
(
d
1
+
d
2
−
2
)
{\displaystyle {\frac {(2d_{1}+d_{2}-2){\sqrt {8(d_{2}-4)}}}{(d_{2}-6){\sqrt {d_{1}(d_{1}+d_{2}-2)}}}}\!}
for d 2 > 6 Excess kurtosis
see text Entropy
ln
Γ
(
d
1
2
)
+
ln
Γ
(
d
2
2
)
−
ln
Γ
(
d
1
+
d
2
2
)
+
{\displaystyle \ln \Gamma \left({\tfrac {d_{1}}{2}}\right)+\ln \Gamma \left({\tfrac {d_{2}}{2}}\right)-\ln \Gamma \left({\tfrac {d_{1}+d_{2}}{2}}\right)+\!}
(
1
−
d
1
2
)
ψ
(
1
+
d
1
2
)
−
(
1
+
d
2
2
)
ψ
(
1
+
d
2
2
)
{\displaystyle \left(1-{\tfrac {d_{1}}{2}}\right)\psi \left(1+{\tfrac {d_{1}}{2}}\right)-\left(1+{\tfrac {d_{2}}{2}}\right)\psi \left(1+{\tfrac {d_{2}}{2}}\right)\!}
+
(
d
1
+
d
2
2
)
ψ
(
d
1
+
d
2
2
)
+
ln
d
1
d
2
{\displaystyle +\left({\tfrac {d_{1}+d_{2}}{2}}\right)\psi \left({\tfrac {d_{1}+d_{2}}{2}}\right)+\ln {\frac {d_{1}}{d_{2}}}\!}
[1] MGF
does not exist, raw moments defined in text and in [2] [3] CF
see text
In
Definition
The F -distribution with d 1 and d 2 degrees of freedom is the distribution of
X
=
S
1
/
d
1
S
2
/
d
2
{\displaystyle X={\frac {S_{1}/d_{1}}{S_{2}/d_{2}}}}
where
S
1
{\textstyle S_{1}}
and
S
2
{\textstyle S_{2}}
are independent
chi-square distributions
with respective degrees of freedom
d
1
{\textstyle d_{1}}
and
d
2
{\textstyle d_{2}}
.
It can be shown to follow that the probability density function (pdf) for X is given by
f
(
x
;
d
1
,
d
2
)
=
(
d
1
x
)
d
1
d
2
d
2
(
d
1
x
+
d
2
)
d
1
+
d
2
x
B
(
d
1
2
,
d
2
2
)
=
1
B
(
d
1
2
,
d
2
2
)
(
d
1
d
2
)
d
1
2
x
d
1
2
−
1
(
1
+
d
1
d
2
x
)
−
d
1
+
d
2
2
{\displaystyle {\begin{aligned}f(x;d_{1},d_{2})&={\frac {\sqrt {\frac {(d_{1}x)^{d_{1}}\,\,d_{2}^{d_{2}}}{(d_{1}x+d_{2})^{d_{1}+d_{2}}}}}{x\operatorname {B} \left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\\[5pt]&={\frac {1}{\operatorname {B} \left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\left({\frac {d_{1}}{d_{2}}}\right)^{\frac {d_{1}}{2}}x^{{\frac {d_{1}}{2}}-1}\left(1+{\frac {d_{1}}{d_{2}}}\,x\right)^{-{\frac {d_{1}+d_{2}}{2}}}\end{aligned}}}
for real x > 0. Here
B
{\displaystyle \mathrm {B} }
is the
positive integers
, but the distribution is well-defined for positive real values of these parameters.
The cumulative distribution function is
F
(
x
;
d
1
,
d
2
)
=
I
d
1
x
/
(
d
1
x
+
d
2
)
(
d
1
2
,
d
2
2
)
,
{\displaystyle F(x;d_{1},d_{2})=I_{d_{1}x/(d_{1}x+d_{2})}\left({\tfrac {d_{1}}{2}},{\tfrac {d_{2}}{2}}\right),}
where I is the
regularized incomplete beta function
.
The expectation, variance, and other details about the F(d 1 , d 2 ) are given in the sidebox; for d 2 > 8, the
excess kurtosis
is
γ
2
=
12
d
1
(
5
d
2
−
22
)
(
d
1
+
d
2
−
2
)
+
(
d
2
−
4
)
(
d
2
−
2
)
2
d
1
(
d
2
−
6
)
(
d
2
−
8
)
(
d
1
+
d
2
−
2
)
.
{\displaystyle \gamma _{2}=12{\frac {d_{1}(5d_{2}-22)(d_{1}+d_{2}-2)+(d_{2}-4)(d_{2}-2)^{2}}{d_{1}(d_{2}-6)(d_{2}-8)(d_{1}+d_{2}-2)}}.}
The k -th moment of an F(d 1 , d 2 ) distribution exists and is finite only when 2k < d 2 and it is equal to
μ
X
(
k
)
=
(
d
2
d
1
)
k
Γ
(
d
1
2
+
k
)
Γ
(
d
1
2
)
Γ
(
d
2
2
−
k
)
Γ
(
d
2
2
)
.
{\displaystyle \mu _{X}(k)=\left({\frac {d_{2}}{d_{1}}}\right)^{k}{\frac {\Gamma \left({\tfrac {d_{1}}{2}}+k\right)}{\Gamma \left({\tfrac {d_{1}}{2}}\right)}}{\frac {\Gamma \left({\tfrac {d_{2}}{2}}-k\right)}{\Gamma \left({\tfrac {d_{2}}{2}}\right)}}.}
[6]
The F -distribution is a particular parametrization of the beta prime distribution , which is also called the beta distribution of the second kind.
The characteristic function is listed incorrectly in many standard references (e.g.,[3] ). The correct expression [7] is
φ
d
1
,
d
2
F
(
s
)
=
Γ
(
d
1
+
d
2
2
)
Γ
(
d
2
2
)
U
(
d
1
2
,
1
−
d
2
2
,
−
d
2
d
1
ı
s
)
{\displaystyle \varphi _{d_{1},d_{2}}^{F}(s)={\frac {\Gamma \left({\frac {d_{1}+d_{2}}{2}}\right)}{\Gamma \left({\tfrac {d_{2}}{2}}\right)}}U\!\left({\frac {d_{1}}{2}},1-{\frac {d_{2}}{2}},-{\frac {d_{2}}{d_{1}}}\imath s\right)}
where U (a , b , z ) is the confluent hypergeometric function of the second kind.
Characterization
A random variate of the F -distribution with parameters
d
1
{\displaystyle d_{1}}
and
d
2
{\displaystyle d_{2}}
arises as the ratio of two appropriately scaled chi-squared variates:[8]
X
=
U
1
/
d
1
U
2
/
d
2
{\displaystyle X={\frac {U_{1}/d_{1}}{U_{2}/d_{2}}}}
where
In instances where the F -distribution is used, for example in the analysis of variance , independence of
U
1
{\displaystyle U_{1}}
and
U
2
{\displaystyle U_{2}}
might be demonstrated by applying Cochran's theorem .
Equivalently, the random variable of the F -distribution may also be written
X
=
s
1
2
σ
1
2
÷
s
2
2
σ
2
2
,
{\displaystyle X={\frac {s_{1}^{2}}{\sigma _{1}^{2}}}\div {\frac {s_{2}^{2}}{\sigma _{2}^{2}}},}
where
s
1
2
=
S
1
2
d
1
{\displaystyle s_{1}^{2}={\frac {S_{1}^{2}}{d_{1}}}}
and
s
2
2
=
S
2
2
d
2
{\displaystyle s_{2}^{2}={\frac {S_{2}^{2}}{d_{2}}}}
,
S
1
2
{\displaystyle S_{1}^{2}}
is the sum of squares of
d
1
{\displaystyle d_{1}}
random variables from normal distribution
N
(
0
,
σ
1
2
)
{\displaystyle N(0,\sigma _{1}^{2})}
and
S
2
2
{\displaystyle S_{2}^{2}}
is the sum of squares of
d
2
{\displaystyle d_{2}}
random variables from normal distribution
N
(
0
,
σ
2
2
)
{\displaystyle N(0,\sigma _{2}^{2})}
.
[discuss ] [citation needed ]
In a
frequentist
context, a scaled
F -distribution therefore gives the probability
p
(
s
1
2
/
s
2
2
∣
σ
1
2
,
σ
2
2
)
{\displaystyle p(s_{1}^{2}/s_{2}^{2}\mid \sigma _{1}^{2},\sigma _{2}^{2})}
, with the
F -distribution itself, without any scaling, applying where
σ
1
2
{\displaystyle \sigma _{1}^{2}}
is being taken equal to
σ
2
2
{\displaystyle \sigma _{2}^{2}}
. This is the context in which the
F -distribution most generally appears in
F -tests: where the null hypothesis is that two independent normal variances are equal, and the observed sums of some appropriately selected squares are then examined to see whether their ratio is significantly incompatible with this null hypothesis.
The quantity
X
{\displaystyle X}
has the same distribution in Bayesian statistics, if an uninformative rescaling-invariant Jeffreys prior is taken for the prior probabilities of
σ
1
2
{\displaystyle \sigma _{1}^{2}}
and
σ
2
2
{\displaystyle \sigma _{2}^{2}}
.[9] In this context, a scaled F -distribution thus gives the posterior probability
p
(
σ
2
2
/
σ
1
2
∣
s
1
2
,
s
2
2
)
{\displaystyle p(\sigma _{2}^{2}/\sigma _{1}^{2}\mid s_{1}^{2},s_{2}^{2})}
, where the observed sums
s
1
2
{\displaystyle s_{1}^{2}}
and
s
2
2
{\displaystyle s_{2}^{2}}
are now taken as known.
Properties and related distributions
If
X
∼
χ
d
1
2
{\displaystyle X\sim \chi _{d_{1}}^{2}}
and
Y
∼
χ
d
2
2
{\displaystyle Y\sim \chi _{d_{2}}^{2}}
(, then
X
/
d
1
Y
/
d
2
∼
F
(
d
1
,
d
2
)
{\displaystyle {\frac {X/d_{1}}{Y/d_{2}}}\sim \mathrm {F} (d_{1},d_{2})}
If
X
k
∼
Γ
(
α
k
,
β
k
)
{\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})\,}
(Gamma distribution ) are independent, then
α
2
β
1
X
1
α
1
β
2
X
2
∼
F
(
2
α
1
,
2
α
2
)
{\displaystyle {\frac {\alpha _{2}\beta _{1}X_{1}}{\alpha _{1}\beta _{2}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})}
If
X
∼
Beta
(
d
1
/
2
,
d
2
/
2
)
{\displaystyle X\sim \operatorname {Beta} (d_{1}/2,d_{2}/2)}
(Beta distribution ) then
d
2
X
d
1
(
1
−
X
)
∼
F
(
d
1
,
d
2
)
{\displaystyle {\frac {d_{2}X}{d_{1}(1-X)}}\sim \operatorname {F} (d_{1},d_{2})}
Equivalently, if
X
∼
F
(
d
1
,
d
2
)
{\displaystyle X\sim F(d_{1},d_{2})}
, then
d
1
X
/
d
2
1
+
d
1
X
/
d
2
∼
Beta
(
d
1
/
2
,
d
2
/
2
)
{\displaystyle {\frac {d_{1}X/d_{2}}{1+d_{1}X/d_{2}}}\sim \operatorname {Beta} (d_{1}/2,d_{2}/2)}
.
If
X
∼
F
(
d
1
,
d
2
)
{\displaystyle X\sim F(d_{1},d_{2})}
, then
d
1
d
2
X
{\displaystyle {\frac {d_{1}}{d_{2}}}X}
has a beta prime distribution :
d
1
d
2
X
∼
β
′
(
d
1
2
,
d
2
2
)
{\displaystyle {\frac {d_{1}}{d_{2}}}X\sim \operatorname {\beta ^{\prime }} \left({\tfrac {d_{1}}{2}},{\tfrac {d_{2}}{2}}\right)}
.
If
X
∼
F
(
d
1
,
d
2
)
{\displaystyle X\sim F(d_{1},d_{2})}
then
Y
=
lim
d
2
→
∞
d
1
X
{\displaystyle Y=\lim _{d_{2}\to \infty }d_{1}X}
has the chi-squared distribution
χ
d
1
2
{\displaystyle \chi _{d_{1}}^{2}}
F
(
d
1
,
d
2
)
{\displaystyle F(d_{1},d_{2})}
is equivalent to the scaled Hotelling's T-squared distribution
d
2
d
1
(
d
1
+
d
2
−
1
)
T
2
(
d
1
,
d
1
+
d
2
−
1
)
{\displaystyle {\frac {d_{2}}{d_{1}(d_{1}+d_{2}-1)}}\operatorname {T} ^{2}(d_{1},d_{1}+d_{2}-1)}
.
If
X
∼
F
(
d
1
,
d
2
)
{\displaystyle X\sim F(d_{1},d_{2})}
then
X
−
1
∼
F
(
d
2
,
d
1
)
{\displaystyle X^{-1}\sim F(d_{2},d_{1})}
.
If
X
∼
t
(
n
)
{\displaystyle X\sim t_{(n)}}
— Student's t-distribution — then:
X
2
∼
F
(
1
,
n
)
X
−
2
∼
F
(
n
,
1
)
{\displaystyle {\begin{aligned}X^{2}&\sim \operatorname {F} (1,n)\\X^{-2}&\sim \operatorname {F} (n,1)\end{aligned}}}
F -distribution is a special case of type 6 Pearson distribution
If
X
{\displaystyle X}
and
Y
{\displaystyle Y}
are independent, with
X
,
Y
∼
{\displaystyle X,Y\sim }
Laplace(μ , b ) then
|
X
−
μ
|
|
Y
−
μ
|
∼
F
(
2
,
2
)
{\displaystyle {\frac {|X-\mu |}{|Y-\mu |}}\sim \operatorname {F} (2,2)}
If
X
∼
F
(
n
,
m
)
{\displaystyle X\sim F(n,m)}
then
log
X
2
∼
FisherZ
(
n
,
m
)
{\displaystyle {\tfrac {\log {X}}{2}}\sim \operatorname {FisherZ} (n,m)}
(Fisher's z-distribution )
The noncentral F -distribution simplifies to the F -distribution if
λ
=
0
{\displaystyle \lambda =0}
.
The doubly noncentral F -distribution simplifies to the F -distribution if
λ
1
=
λ
2
=
0
{\displaystyle \lambda _{1}=\lambda _{2}=0}
If
Q
X
(
p
)
{\displaystyle \operatorname {Q} _{X}(p)}
is the quantile p for
X
∼
F
(
d
1
,
d
2
)
{\displaystyle X\sim F(d_{1},d_{2})}
and
Q
Y
(
1
−
p
)
{\displaystyle \operatorname {Q} _{Y}(1-p)}
is the quantile
1
−
p
{\displaystyle 1-p}
for
Y
∼
F
(
d
2
,
d
1
)
{\displaystyle Y\sim F(d_{2},d_{1})}
, then
Q
X
(
p
)
=
1
Q
Y
(
1
−
p
)
.
{\displaystyle \operatorname {Q} _{X}(p)={\frac {1}{\operatorname {Q} _{Y}(1-p)}}.}
F -distribution is an instance of ratio distributions
W -distribution[10] is a unique parametrization of F-distribution.
See also
References
External links
Discrete univariate
with finite support with infinite support
Continuous univariate
supported on a bounded interval supported on a semi-infinite interval supported on the whole real line with support whose type varies
Mixed univariate
Multivariate (joint) Directional Degenerate and singular Families