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Raised cosine
Probability density function
Cumulative distribution function
Parameters
μ
{\displaystyle \mu \,}
(
real )
s
>
0
{\displaystyle s>0\,}
(
real )
Support
x
∈
μ
−
s
,
μ
+
s
{\displaystyle x\in [\mu -s,\mu +s]\,}
PDF
1
2
s
1
+
cos
(
x
−
μ
s
π
)
=
1
s
hvc
(
x
−
μ
s
π
)
{\displaystyle {\frac {1}{2s}}\left[1+\cos \left({\frac {x-\mu }{s}}\,\pi \right)\right]\,={\frac {1}{s}}\operatorname {hvc} \left({\frac {x-\mu }{s}}\,\pi \right)\,}
CDF
1
2
1
+
x
−
μ
s
+
1
π
sin
(
x
−
μ
s
π
)
{\displaystyle {\frac {1}{2}}\left[1+{\frac {x-\mu }{s}}+{\frac {1}{\pi }}\sin \left({\frac {x-\mu }{s}}\,\pi \right)\right]}
Mean
μ
{\displaystyle \mu \,}
Median
μ
{\displaystyle \mu \,}
Mode
μ
{\displaystyle \mu \,}
Variance
s
2
(
1
3
−
2
π
2
)
{\displaystyle s^{2}\left({\frac {1}{3}}-{\frac {2}{\pi ^{2}}}\right)\,}
Skewness
0
{\displaystyle 0\,}
Excess kurtosis
6
(
90
−
π
4
)
5
(
π
2
−
6
)
2
=
−
0.59376
…
{\displaystyle {\frac {6(90-\pi ^{4})}{5(\pi ^{2}-6)^{2}}}=-0.59376\ldots \,}
MGF
π
2
sinh
(
s
t
)
s
t
(
π
2
+
s
2
t
2
)
e
μ
t
{\displaystyle {\frac {\pi ^{2}\sinh(st)}{st(\pi ^{2}+s^{2}t^{2})}}\,e^{\mu t}}
CF
π
2
sin
(
s
t
)
s
t
(
π
2
−
s
2
t
2
)
e
i
μ
t
{\displaystyle {\frac {\pi ^{2}\sin(st)}{st(\pi ^{2}-s^{2}t^{2})}}\,e^{i\mu t}}
In
probability theory and
statistics , the raised cosine distribution is a continuous
probability distribution
supported on the interval
μ
−
s
,
μ
+
s
{\displaystyle [\mu -s,\mu +s]}
. The
probability density function (PDF) is
f
(
x
;
μ
,
s
)
=
1
2
s
1
+
cos
(
x
−
μ
s
π
)
=
1
s
hvc
(
x
−
μ
s
π
)
for
μ
−
s
≤
x
≤
μ
+
s
{\displaystyle f(x;\mu ,s)={\frac {1}{2s}}\left[1+\cos \left({\frac {x-\mu }{s}}\,\pi \right)\right]\,={\frac {1}{s}}\operatorname {hvc} \left({\frac {x-\mu }{s}}\,\pi \right){\text{ for }}\mu -s\leq x\leq \mu +s}
and zero otherwise. The cumulative distribution function (CDF) is
F
(
x
;
μ
,
s
)
=
1
2
1
+
x
−
μ
s
+
1
π
sin
(
x
−
μ
s
π
)
{\displaystyle F(x;\mu ,s)={\frac {1}{2}}\left[1+{\frac {x-\mu }{s}}+{\frac {1}{\pi }}\sin \left({\frac {x-\mu }{s}}\,\pi \right)\right]}
for
μ
−
s
≤
x
≤
μ
+
s
{\displaystyle \mu -s\leq x\leq \mu +s}
and zero for
x
<
μ
−
s
{\displaystyle x<\mu -s}
and unity for
x
>
μ
+
s
{\displaystyle x>\mu +s}
.
The
moments of the raised cosine distribution are somewhat complicated in the general case, but are considerably simplified for the standard raised cosine distribution. The standard raised cosine distribution is just the raised cosine distribution with
μ
=
0
{\displaystyle \mu =0}
and
s
=
1
{\displaystyle s=1}
. Because the standard raised cosine distribution is an
even function , the odd moments are zero. The even moments are given by:
E
(
x
2
n
)
=
1
2
∫
−
1
1
1
+
cos
(
x
π
)
x
2
n
d
x
=
∫
−
1
1
x
2
n
hvc
(
x
π
)
d
x
=
1
n
+
1
+
1
1
+
2
n
1
F
2
(
n
+
1
2
;
1
2
,
n
+
3
2
;
−
π
2
4
)
{\displaystyle {\begin{aligned}\operatorname {E} (x^{2n})&={\frac {1}{2}}\int _{-1}^{1}[1+\cos(x\pi )]x^{2n}\,dx=\int _{-1}^{1}x^{2n}\operatorname {hvc} (x\pi )\,dx\\[5pt]&={\frac {1}{n+1}}+{\frac {1}{1+2n}}\,_{1}F_{2}\left(n+{\frac {1}{2}};{\frac {1}{2}},n+{\frac {3}{2}};{\frac {-\pi ^{2}}{4}}\right)\end{aligned}}}
where
1
F
2
{\displaystyle \,_{1}F_{2}}
is a
generalized hypergeometric function .
See also
References
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