marginal probability, probability,

Question

A mosquito is trying to annoy you. At any given time, it can be found anywhere within a radius R from your center (assume that it can be found inside you). Use spherical coordinate system. i. Determine the total volume that the mosquito can be found in? Hint: Assume that he can be anywhere in a sphere of radius R from your center. ii. If it is equally probable to find the mosquito at any position throughout the sphere, we can assign a uniform probability density for the mosquito’s position: p(r,θ,Ф) = C , where C is a constant. Find C so that the probability is properly normalized. iii. What is the probability for the mosquito to be a distance r from your center, regardless of its angular position? Does your answer make sense? Hint: You will need to marginalize over the parameters θ and Ф. Recall that the infinitesimal lengths are dr, rdθ, rsinθdФ in spherical coordinates. iv. You don’t mind the mosquito as long as you can see him. In that case he is in the octant of the sphere above your center and in front of your face. This octant spans 90º in θ and 90º in Ф. What is the probability that you will find the mosquito there? v. What is the probability that you won’t be able to see it? That is when it is not in the visible octant? vi. What is the probability that the mosquito will be at any angle θ irrespective of r or Ф? vii. You really don’t want to be bit on the head. This means that you don’t want the mosquito to have a position where 0≤θ≤π/4 . What is p(0≤θ≤π/4)?