Simulating Particle Hits on an Infinite Screen

Question

# Interview Question: Simulating Particle Hits on an Infinite Screen

You are given a simple physical scattering model.

A point source is located at the origin `(0, 0)`. It emits particles toward an **infinite vertical screen** placed at position `x = 1`. Each particle is emitted at an angle `theta` relative to the positive x-axis. Assume:

- `theta` is sampled **uniformly** from the interval `[-pi/2, pi/2]`
- each particle travels in a straight line until it hits the screen
- the screen is infinite in the y-direction

For a given particle emitted at angle `theta`, its trajectory intersects the screen at some vertical position `y`.

## Part 1: Derivation

1. Derive the formula for the hit position `y` on the screen as a function of `theta`.
2. Using the fact that `theta ~ Uniform[-pi/2, pi/2]`, determine the theoretical distribution of `y`.
3. Write down the exact probability density function (PDF) of `y`.

## Part 2: Simulation

Implement a simulation that:

1. draws `N` independent samples of `theta`
2. converts them into screen hit positions `y`
3. plots a histogram of the simulated `y` values
4. overlays the theoretical PDF on top of the histogram to verify correctness

Use a

Accepted Answer

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