Openai·Software Engineer·Technical Phone Screen·Senior

SeniorPrefer not to say

Jun 2026Remote

Summary

OpenAI software engineer interview that went deep into probability theory and numerical simulation. Not your typical coding round. They wanted both the math derivation and working code to verify it, which I was not fully prepared for.

Questions Asked (2)

Model a point light source emitting particles at uniformly random angles toward an infinite vertical screen. Derive mathematically why the x-coordinates where particles hit the screen follow a Cauchy distribution.

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

The setup sounds like a physics problem but it's really a probability derivation.

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AI HintsAI Generated

Suggested Approach

Set up the geometry with the light source at the origin and the screen as a vertical line at distance d. Express the x-coordinate of impact as d * tan(θ), where θ is the angle from the perpendicular. Since θ is uniform, derive the probability density function of X using the transformation of variables, showing it matches the Cauchy distribution.

Pro tip: Emphasize that the heavy tails of the Cauchy distribution arise because angles near ±π/2 produce extremely large x-coordinates, and mention that the mean and variance are undefined, which is a key property.

1. Define the geometry and variables

Place the point light source at the origin and the infinite vertical screen at x = d (or y = d). Let θ be the angle of emission relative to the perpendicular, uniformly distributed over (-π/2, π/2).

2. Relate x-coordinate to angle

Using trigonometry, the x-coordinate of impact is X = d * tan(θ). Note that θ is uniform, so its probability density function is constant over the interval.

3. Apply transformation of variables

Use the formula for the density of a transformed random variable: f_X(x) = f_θ(θ) * |dθ/dx|. Compute the derivative dθ/dx = d / (d^2 + x^2).

4. Derive the Cauchy density

Substitute the uniform density and the derivative to obtain f_X(x) = 1/(π d) * 1/(1 + (x/d)^2), which is the Cauchy distribution with location 0 and scale d.

5. Discuss implications and properties

Highlight that the Cauchy distribution has heavy tails, no defined mean or variance, and that this arises because angles near ±π/2 map to very large |x| values.

Key Points to Mention

Uniform distribution of angles over (-π/2, π/2)
Geometric relationship: X = d * tan(θ)
Transformation of variables technique for probability densities
Derivative dθ/dx = d / (d^2 + x^2)
Resulting density: f_X(x) = 1/(π d) * 1/(1 + (x/d)^2)
Cauchy distribution's heavy tails and undefined mean/variance

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.

Now simulate the particle emission process numerically: sample uniform random angles, compute the screen impact x-coordinates, and verify your derivation by comparing a histogram of the simulated values against the theoretical Cauchy PDF.

A/B Testing & ExperimentationTechnical Trade-offs

Author's notes

This part I actually handled better.

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AI HintsAI Generated

Suggested Approach

First, clearly state the mathematical derivation: for a point source emitting particles uniformly in angle, the impact x-coordinate on a screen follows a Cauchy distribution. Then, outline a numerical simulation plan: sample angles uniformly, compute x = d * tan(θ), and generate a histogram. Finally, compare the histogram to the theoretical Cauchy PDF, using appropriate visualization and statistical measures to verify agreement.

Pro tip: Mention that the Cauchy distribution has heavy tails, so the sample mean and variance are undefined; use robust statistics like median and interquartile range for comparison, and consider log-scale plots to visualize the tails.

1. Derive the theoretical distribution

Show that if θ ~ Uniform(-π/2, π/2), then x = d * tan(θ) follows a Cauchy distribution with location 0 and scale d. Derive the PDF using the transformation of variables.

2. Design the simulation

Choose the number of samples N (e.g., 10^5), the screen distance d, and the random number generator. Sample angles uniformly from (-π/2, π/2) and compute x = d * tan(θ).

3. Generate and visualize the histogram

Plot a histogram of the simulated x values with appropriate binning. Overlay the theoretical Cauchy PDF for comparison. Use a log-scale on the y-axis to better see the tails.

4. Quantitatively compare

Compute summary statistics (e.g., median, interquartile range) and perform a Kolmogorov-Smirnov test or compare quantiles to assess the goodness of fit.

5. Discuss limitations and extensions

Acknowledge that the Cauchy distribution has undefined mean and variance, so avoid using those. Suggest extensions like varying d, adding noise, or considering finite screen size.

Key Points to Mention

Uniform angle sampling: θ ~ Uniform(-π/2, π/2) to cover all possible directions.
Transformation: x = d * tan(θ) where d is the distance to the screen.
Cauchy PDF: f(x) = 1/(π d (1 + (x/d)^2)).
Heavy tails: Cauchy has no finite mean or variance, so use median and IQR.
Visualization: histogram with log-scale y-axis and overlay of theoretical PDF.
Statistical test: Kolmogorov-Smirnov test for goodness of fit.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.