Question

probability problem
Consider the variable y defined as

look at attachment

Answer 1

The sum of all probabilities given as a percentage equals one.

In a discrete probability distribution
\[\sum\limits_{n=1}^\infty P(n)=1\qquad\qquad n=\{1,2,3,...\}\]
In a continuos probability distribution
\[\int\limits_{-\infty}^\infty P(x)\text dx=1\qquad\qquad-\infty\leq x\leq\infty\]
For distribution all the way around a point
\[\int\limits_{0}^{2\pi} P(\vartheta)\text d\vartheta =1\qquad\qquad0\leq\vartheta\leq 2\pi\]

Answer 2

In this question the probability distribution has an arc of pi
\[\int\limits_{0}^{\pi} P(\theta)\text d \theta=1\qquad\qquad0\leq\theta\leq \pi\]
and every value has the same probability implies \( P(\theta)\) is a constant
\[\int\limits_{0}^{\pi}
\text d\theta=\frac{1}{P(\theta)}\]\[\left.\theta\right|_0^\pi=\frac{1}{P(\theta)}\]\[\pi=\frac1{P(\theta)}\]\[P(\theta)=\pi^{-1}\]