Normal to Standard CDF Transform

This question asks you to rewrite an improper integral with a quadratic exponent in terms of the standard normal cumulative distribution function. The setup is a generic exponential of a quadratic polynomial integrated from minus infinity up to a variable limit. The key challenge is to recognize how to recast that expression into something that looks like a Gaussian tail probability, so that it matches the standard normal CDF up to scaling and shifting. This style of problem appears in mathematically heavy quant roles and in theory-focused interviews where facility with Gaussian transformations is essential.

To tackle it, you need comfort with completing the square in the exponent, manipulating affine changes of variables, and handling constants that arise from the transformation. Conceptually, it leans on seeing any quadratic exponent as a distorted normal density, then undoing that distortion to recover the standard form. An interviewer is watching for algebraic precision, correct control of signs and scaling factors, and a clear understanding of how analytic integrals connect to probability distributions rather than rote memorization of special-function formulas.

What it tests

Whenever you encounter an integral of the form $\int e^{A t^2 + B t + C} dt$, the key is to recognize that the exponent is a quadratic in $t$, which can always be rewritten by completing the square. This transformation allows you to express the original function in terms of a shifted and scaled version of the standard normal density or cumulative distribution function. The reason this works is that the Gaussian (normal) distribution is defined by a quadratic exponent, so any quadratic exponent can be mapped to the standard normal form via an affine change of variables. This is a powerful unifying structure: all such integrals are, up to scaling and shifting, just rescaled versions of the normal CDF or error function. The principle holds because the set of all quadratic functions is closed under completion of the square, and the normal distribution is the canonical example of such a function in probability and statistics.