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U-quadratic
Probability density function
Parameters
a
:
a
∈
(
−
∞
,
∞
)
{\displaystyle a:~a\in (-\infty ,\infty )}
b
:
b
∈
(
a
,
∞
)
{\displaystyle b:~b\in (a,\infty )}
or
α
:
α
∈
(
0
,
∞
)
{\displaystyle \alpha :~\alpha \in (0,\infty )}
β
:
β
∈
(
−
∞
,
∞
)
,
{\displaystyle \beta :~\beta \in (-\infty ,\infty ),}
Support
x
∈
a
,
b
{\displaystyle x\in [a,b]\!}
PDF
α
(
x
−
β
)
2
{\displaystyle \alpha \left(x-\beta \right)^{2}}
CDF
α
3
(
(
x
−
β
)
3
+
(
β
−
a
)
3
)
{\displaystyle {\alpha \over 3}\left((x-\beta )^{3}+(\beta -a)^{3}\right)}
Mean
a
+
b
2
{\displaystyle {a+b \over 2}}
Median
a
+
b
2
{\displaystyle {a+b \over 2}}
Mode
a
and
b
{\displaystyle a{\text{ and }}b}
Variance
3
20
(
b
−
a
)
2
{\displaystyle {3 \over 20}(b-a)^{2}}
Skewness
0
{\displaystyle 0}
Excess kurtosis
3
112
(
b
−
a
)
4
{\displaystyle {3 \over 112}(b-a)^{4}}
Entropy
TBD
MGF
See text
CF
See text
In
probability theory and
statistics , the U-quadratic distribution is a continuous
probability distribution defined by a unique
convex quadratic function with lower limit a and upper limit b .
f
(
x
|
a
,
b
,
α
,
β
)
=
α
(
x
−
β
)
2
,
for
x
∈
a
,
b
.
{\displaystyle f(x|a,b,\alpha ,\beta )=\alpha \left(x-\beta \right)^{2},\quad {\text{for }}x\in [a,b].}
Parameter relations
This distribution has effectively only two parameters a , b , as the other two are explicit functions of the support defined by the former two parameters:
β
=
b
+
a
2
{\displaystyle \beta ={b+a \over 2}}
(gravitational balance center, offset), and
α
=
12
(
b
−
a
)
3
{\displaystyle \alpha ={12 \over \left(b-a\right)^{3}}}
(vertical scale).
Related distributions
One can introduce a vertically inverted (
∩
{\displaystyle \cap }
)-quadratic distribution in analogous fashion.
Applications
This distribution is a useful model for symmetric
bimodal processes. Other continuous distributions allow more flexibility, in terms of relaxing the symmetry and the quadratic shape of the density function, which are enforced in the U-quadratic distribution – e.g.,
beta distribution and
gamma distribution .
Moment generating function
M
X
(
t
)
=
−
3
(
e
a
t
(
4
+
(
a
2
+
2
a
(
−
2
+
b
)
+
b
2
)
t
)
−
e
b
t
(
4
+
(
−
4
b
+
(
a
+
b
)
2
)
t
)
)
(
a
−
b
)
3
t
2
{\displaystyle M_{X}(t)={-3\left(e^{at}(4+(a^{2}+2a(-2+b)+b^{2})t)-e^{bt}(4+(-4b+(a+b)^{2})t)\right) \over (a-b)^{3}t^{2}}}
Characteristic function
ϕ
X
(
t
)
=
3
i
(
e
i
a
t
e
i
b
t
(
4
i
−
(
−
4
b
+
(
a
+
b
)
2
)
t
)
)
(
a
−
b
)
3
t
2
{\displaystyle \phi _{X}(t)={3i\left(e^{iate^{ibt}}(4i-(-4b+(a+b)^{2})t)\right) \over (a-b)^{3}t^{2}}}
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