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UnkleRhaukus
Group Title
find \[\langle \theta \rangle \]
for \(0≤\theta≤\pi/2\)
 one year ago
 one year ago
UnkleRhaukus Group Title
find \[\langle \theta \rangle \] for \(0≤\theta≤\pi/2\)
 one year ago
 one year ago

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UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
\[\langle x\rangle=\int\limits_0^{\pi/2}\theta\cdot\rho(\theta)\cdot\text d\theta \]
 one year ago

hartnn Group TitleBest ResponseYou've already chosen the best response.4
whats \(\rho(\theta)\) ?
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
probability density
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
\[1=\int\limits_0^{\pi/2}\rho(\theta)\cdot\text d\theta\]
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
assuming a unifirm probability distribution \[\rho(\theta)=A\] \[\frac 1A=\int\limits_0^{\pi/2}\text d\theta\]
 one year ago

hartnn Group TitleBest ResponseYou've already chosen the best response.4
A=2/\(\pi\) then
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
\[\frac 1A=\theta_0^{\pi/2}=\frac\pi2\] \[A=\frac 2\pi\]
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
so, \[\langle x\rangle=\frac 2\pi\int\limits_0^{\pi/2}\theta\cdot\text d\theta\]
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
\[=\frac 2\pi \frac{\theta^2}{2} _0^{\pi/2}\] \[=\frac 2\pi \frac{\left(\pi/2\right)^2}{2}\] \[=\frac\pi 4\]
 one year ago

hartnn Group TitleBest ResponseYou've already chosen the best response.4
nice, pdf was assumed or given? assuming it as uniform simplified it a lot.....
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
i probably should have stated uniform probability distribution in the question
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
so for \(0≤\theta≤2\pi\)\[\langle \theta \rangle =\pi\] the expected angle in a circle is 180°
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
makes sense
 one year ago

hartnn Group TitleBest ResponseYou've already chosen the best response.4
uniform pdf will always give you midpoint as expectation....
 one year ago

UnkleRhaukus Group TitleBest ResponseYou've already chosen the best response.0
right
 one year ago
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