Linear Combination of Normals

What this linear combination of normals interview question tests

This is a medium-difficulty probability question that asks you to work with sums and differences of independent normal random variables. It assesses whether you can manipulate linear combinations of Gaussians and reduce a joint probability statement to a single standard-normal cumulative probability.

The core skill is recognizing that linear combinations of independent normals are themselves normal, then computing the mean and variance of the resulting distribution. Once you have reduced the problem to a single normal random variable, you standardize it and express the answer in terms of the standard-normal cumulative distribution function (often written as Phi). Interviewers value clean algebra and clear notation over speed.

• Independence and the sum of variances
• Linearity of expectation for combinations of random variables
• Standardizing a normal to N(0,1) form
• Interpreting cumulative distribution function notation