To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

# Raised cosine distribution

Parameters Probability density function Cumulative distribution function ${\displaystyle \mu \,}$(real) ${\displaystyle s>0\,}$(real) ${\displaystyle x\in [\mu -s,\mu +s]\,}$ ${\displaystyle {\frac {1}{2s}}\left[1+\cos \left({\frac {x-\mu }{s}}\,\pi \right)\right]\,={\frac {1}{s}}\operatorname {hvc} \left({\frac {x-\mu }{s}}\,\pi \right)\,}$ ${\displaystyle {\frac {1}{2}}\left[1+{\frac {x-\mu }{s}}+{\frac {1}{\pi }}\sin \left({\frac {x-\mu }{s}}\,\pi \right)\right]}$ ${\displaystyle \mu \,}$ ${\displaystyle \mu \,}$ ${\displaystyle \mu \,}$ ${\displaystyle s^{2}\left({\frac {1}{3}}-{\frac {2}{\pi ^{2}}}\right)\,}$ ${\displaystyle 0\,}$ ${\displaystyle {\frac {6(90-\pi ^{4})}{5(\pi ^{2}-6)^{2}}}\,}$ ${\displaystyle {\frac {\pi ^{2}\sinh(st)}{st(\pi ^{2}+s^{2}t^{2})}}\,e^{\mu t}}$ ${\displaystyle {\frac {\pi ^{2}\sin(st)}{st(\pi ^{2}-s^{2}t^{2})}}\,e^{i\mu t}}$

In probability theory and statistics, the raised cosine distribution is a continuous probability distribution supported on the interval ${\displaystyle [\mu -s,\mu +s]}$. The probability density function (PDF) is

${\displaystyle f(x;\mu ,s)={\frac {1}{2s}}\left[1+\cos \left({\frac {x-\mu }{s}}\,\pi \right)\right]\,={\frac {1}{s}}\operatorname {hvc} \left({\frac {x-\mu }{s}}\,\pi \right)\,}$

for ${\displaystyle \mu -s\leq x\leq \mu +s}$ and zero otherwise. The cumulative distribution function (CDF) is

${\displaystyle F(x;\mu ,s)={\frac {1}{2}}\left[1+{\frac {x-\mu }{s}}+{\frac {1}{\pi }}\sin \left({\frac {x-\mu }{s}}\,\pi \right)\right]}$

for ${\displaystyle \mu -s\leq x\leq \mu +s}$ and zero for ${\displaystyle x<\mu -s}$ and unity for ${\displaystyle x>\mu +s}$.

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 ${\displaystyle \mu =0}$ and ${\displaystyle s=1}$. Because the standard raised cosine distribution is an even function, the odd moments are zero. The even moments are given by:

{\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 ${\displaystyle \,_{1}F_{2}}$ is a generalized hypergeometric function.

• 1/5
Views:
56 688
1 326
10 150
1 303
411 936
• ✪ 9. Multiple Continuous Random Variables
• ✪ Cosine Rule (2 of 2: Example questions)
• ✪ 21. Ideal Quantum Gases Part 1
• ✪ 📓 Differentiation: Power Rule Binomial Theorem examples problems (calculus)
• ✪ 3. The Wave Function