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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]\,}
Unknown type
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 }
Unknown type
s
2
(
1
3
−
2
π
2
)
{\displaystyle s^{2}\left({\frac {1}{3}}-{\frac {2}{\pi ^{2}}}\right)\,}
Skewness
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}}
Në teorinë e probabilitetit dhe statistikë , shpërndarja e ngritur e kosinusit është një shpërndarje e vazhdueshme probabiliteti e mbështetur në intervalin
[
μ
−
s
,
μ
+
s
]
{\displaystyle [\mu -s,\mu +s]}
Funksioni i dendësisë së probabilitetit (PDF) është
f
(
x
;
μ
,
s
)
=
1
2
s
[
1
+
cos
(
x
−
μ
s
π
)
]
=
1
s
hvc
(
x
−
μ
s
π
)
{\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)\,}
për
μ
−
s
≤
x
≤
μ
+
s
{\displaystyle \mu -s\leq x\leq \mu +s}
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]}
për
μ
−
s
≤
x
≤
μ
+
s
{\displaystyle \mu -s\leq x\leq \mu +s}
x
<
μ
−
s
{\displaystyle x<\mu -s}
x
>
μ
+
s
{\displaystyle x>\mu +s}
Momentet e shpërndarjes së ngritur të kosinusit janë disi të të ndërlikuara në rastin e përgjithshëm, por janë thjeshtuar ndjeshëm për shpërndarjen standarde të kosinusit të ngritur. Shpërndarja standarde e kosinusit të ngritur është vetëm shpërndarja e kosinusit të ngritur me
μ
=
0
{\displaystyle \mu =0}
s
=
1
{\displaystyle s=1}
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}}}
ku
1
F
2
{\displaystyle \,_{1}F_{2}}
funksion hipergjeometrik i përgjithësuar .
![{\displaystyle x\in [\mu -s,\mu +s]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/021cb61824dc30c9ce4228710410d45d7b8ea2dd)
![{\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)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d8fe6565ff842d25cf9ac9946e3454f278992d8)
![{\displaystyle {\frac {1}{2}}\left[1+{\frac {x-\mu }{s}}+{\frac {1}{\pi }}\sin \left({\frac {x-\mu }{s}}\,\pi \right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a5fe6b908cecf264d0bc4a34c554b027ad3bb88)
![{\displaystyle [\mu -s,\mu +s]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4092914759a1beeec30258141ccc43c0686f8459)
![{\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)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1332aa518f41cc75f685cbc018050732d4971473)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d946fb3c3452f89b48341393ced089a0699fdffd)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd88646853daa97101c07fa637ef17568602b698)
