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

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

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

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

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

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

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

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

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

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

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

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

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