Model particle hits on a screen

A point-like source sits at the origin $(0,0)$. It emits particles into the half-plane to the right, where an unbounded vertical detector plane is positioned at $x = 1$. Each particle’s direction is chosen independently: the angle $\Theta$ is measured from the positive $x$-axis and is uniformly distributed on $[-\pi/2, \pi/2]$. Once a direction is sampled, the particle travels along a straight ray until it reaches the detector. Angles exactly parallel to the detector, $\Theta = \pm \pi/2$, occur with probability zero and may be ignored.

1. Write the vertical coordinate $Y$ of the intersection point as a function of $\Theta$.

2. Using the distribution of $\Theta$, derive the cumulative distribution function $F_Y(y)$ and the probability density function $f_Y(y)$ of $Y$. State the support of $Y$.

3. Implement a Monte Carlo simulation that draws many independent angles, computes the corresponding values of $Y$, and plots a normalized histogram of the hit positions. Overlay the theoretical density from part 2 to verify the derivation.

4. Discuss numerical or visualization effects that could complicate the comparison, such as sample size, histogram bin width, and the presence of very large values of $Y$.

Examples

Example

Input:
1
0
Output: 0.5 0.318309886184