1. The problem statement, all variables and given/known data A rifle shooter aims at a target at a distance D, but has an accuracy probability density ρ(φ)=1/(2Φ) φ∈(-Φ,Φ) where φ is the angle achieved and is bounded by the small angle Φ Part A find the probability density for where the bullet strikes the target , ρ(x) . The target has a width of 2d . H denotes the event where the bullet hits the target whereas M denotes that when it misses. Calculate and depict the probability of hitting P(H;Φ) as a function of Φ for fixed d and D with d=Dtanθ Part B as well as the intrinsic accuracy, the shooter can also set his sights, described by the angle ψ, with respect to the original zero angle . Find the modified probability P(H;Φ;ψ). Part C if ψ is also randomly distributed with probability density ρ(ψ)=1/(2Ψ) ψ∈(-Ψ,Ψ) Part D By considering N shots between resighting, explain how to test whether a shooter is limited by the accuracy or by accuracy of sighting 2. The attempt at a solution Part A ρ(φ)dφ=ρ(x)dx ; x=Dtanφ ; dx=Dsec(φ)2 Therefore, ρ(x)=(1/2Φ)D/(D2+x2) For θ ≤ Φ P(H;Φ) = 1 P(H;Φ)=1-P(M;Φ)=1-2*(θ∫Φρ(φ)dφ) = θ/Φ Part B For Φ>ψ, θ<abs(Φ-ψ) :P(H;Φ;ψ)=1-P(M;Φ;ψ)=1-(θ∫Φ+ψ+-(Φ-ψ)∫-θ)ρ(φ)dφ abs(Φ-ψ)<θ<(Φ+ψ) : P(H;Φ;ψ)=1-P(M;Φ;ψ)=1-θ∫Φ+ψρ(φ)dφ (Φ+ψ)<θ :P(H;Φ;ψ)=1 For Φ<ψ, θ<abs(Φ-ψ) :P(H;Φ;ψ)=0 abs(Φ-ψ)<θ<(Φ+ψ) : P(H;Φ;ψ)=1-P(M;Φ;ψ)=1-θ∫Φ+ψρ(φ)dφ (Φ+ψ)<θ :P(H;Φ;ψ)=1 Part C This is where I start to get stuck, as I have no idea how I could go about combining the two probability density functions to give a distribution function. Looking up wiki, I notice that it might involve something named ' convolution', which I have no knowledge of it whatsoever, and I can't seem to find any alternative ways. Part D This is where I get almost completely clueless as I don't really understand what I'm being asked to do. I suspect it might involve something like comparing the size of two probability distributions limited by different factors (accuracy or accuracy of sighting) . Any help would be greatly appreciated.