Optimal Zero Distance Selection under Stochastic Engagement Range

1. Background

When a rifle is zeroed at distance $z$, the bullet's line of flight intersects the line of sight at that range. At any other target distance $d$, the impact offset is a deterministic function $f(d; z)$ measured in inches relative to the point of aim.

The practical issue is that engagement distance is not known in advance. We therefore model distance as a random variable $D$, and treat zero distance as a decision variable chosen under uncertainty.

2. Notation

3. Engagement Model

The engagement distance follows a Gamma distribution:

Gamma works here because support is positive, the shape can be right-skewed, and the first two moments map directly to operational assumptions.

4. Ballistic Response

For each candidate zero distance, the model computes a trajectory table and linearly interpolates offset at half-yard resolution.

That offset function is then weighted by the engagement density.

5. Objectives

Weighted Mean Offset

This shows signed bias, but positive and negative misses can cancel each other out.

Weighted Variance

This measures consistency across likely engagement ranges.

Weighted Mean Absolute Offset

This is usually the most operational metric when no hold correction is applied, because misses do not cancel by sign.

6. Optimization

These criteria emphasize different behavior: consistency versus average miss distance.

7. Numerical Method

The app evaluates a dense grid on $[0,150]$, interpolates offset values between whole-yard samples, normalizes the Gamma weights, and scores every zero from 1 to 100 yards.

The dashboard exposes riser height, Gamma mean, and Gamma standard deviation as the primary inputs. The recommendation and charts update from those parameters in real time.

8. Baseline Result

Baseline parameters: $\mu = 15$ yd, standard deviation $= 10$ yd, riser $= 2.26$ in.