Question

Continuous probability distribution over an angle range. (Q. in comments)

Answer 1

find the probability density of θ
\[0<\theta<\pi , \rho(\theta)=k\]

\[\int\limits_{0}^{π}\rho(\theta)d \theta=1 \] sum of probabilities is 1

\[ \int\limits_{0}^{π}d \theta=1/k\]

\[\theta]π,0 = 1/k\]
\[π=1/k\] \[k=1/π\]

:.\[\rho(\theta)=(1/π), 0<\theta<\pi\]

Answer 2

the question i have is how would i go about finding the probability distribution for the projection onto the x-axis
\[\rho(x)= ?\]
dx=?

Answer 3

x=rcos(θ)
dx=-rsin(θ)dθ