Linear Prediction Confidence Region Flare-Out

John Cook explains why the confidence region of a tracked object flares out instead of looking conical (or some other shape):

Suppose you’re tracking some object based on its initial position x₀ and initial velocity v₀. The initial position and initial velocity are estimated from normal distributions with standard deviations σ_x and σ_v. (To keep things simple, let’s assume our object is moving in only one dimension and that the distributions around initial position and velocity are independent.)

The confidence region for the object flares out over time, something like the bell of a trumpet.

Read on for the explanation.