- #1

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## Homework Statement

A rifle shooter aims at a target at a distance D, but has an accuracy probability density

ρ(φ)=1/(2

**Φ**) 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/(2

**Ψ**) 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)=(1/2

**Φ**)D/(D

^{2}+x

^{2})

For θ ≤

**Φ**

P(H;

**Φ**) = 1

P(H;

**Φ**)=1-P(M;

**Φ**)=1-2*(

_{θ}∫

^{Φ}ρ(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+ψ}ρ(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+ψ}ρ(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.