P(Y > 4X) for Independent Gaussians

This quant interview question is about working with continuous random variables, specifically independent Gaussian variables, and understanding how inequalities between them translate into probabilities. It appears in quant prep because it forces you to interpret a geometric condition in the plane as a statement about a single random variable. You need to recognize how joint distributions and linear relationships interact in a clean, symmetric setting.

It trains your understanding of linear combinations of independent normals, properties of the normal distribution, and symmetry around the mean. It reinforces fluency with distributional transformations, standardization, and thinking probabilistically about inequalities, all central tools in quantitative finance interview questions.

This matters for quant interviews because pricing models, risk calculations, and PnL distributions often rely on Gaussian assumptions and linear combinations of factors. Interviewers want to see that you can quickly turn an informal probabilistic statement into a precise calculation, a core skill for trading, research, and risk roles.

What it tests

Whenever you have independent normal random variables, any linear combination of them is also normally distributed. The key property is that the resulting normal distribution is fully determined by the linear combination's mean (the sum of the means, each weighted by their coefficients) and variance (the sum of the variances, each weighted by the square of their coefficients, since independence means covariances are zero). For any normal random variable, the probability that it exceeds its mean is always one-half, due to the symmetry of the normal distribution about its mean. This symmetry holds regardless of the coefficients in the linear combination, as long as the mean is zero. Thus, questions about the probability that one normal variable exceeds a linear function of another often reduce to evaluating the sign of a normal variable centered at zero.