Conditional Variance Interview Questions
Conditional variance interview prep for variance after information, case splits, law of total variance intuition, and common mistakes.
Advanced candidates who already know conditional expectation and variance basics.
What conditional variance measures
Conditional variance is the spread of a random variable after some information is known. It is not the original variance with a new label; the conditioning changes the universe being measured.
Start from the conditional mean
A useful definition is the expected squared deviation from the conditional expectation. First find the average under the condition, then measure spread around that updated average.
Concrete example
If a fair die is known to be at least 4, the possible values are 4, 5, and 6. The conditional mean is 5, and the conditional variance is the average of 1, 0, and 1, which is 2/3.
Law of total variance intuition
Overall variance can be split into average within-case variance plus variance between case means. This is the variance version of splitting a problem by hidden states.
When it appears in interviews
Conditional variance can appear after a signal, reveal, filter, or first-stage outcome. State what is known, recompute the conditioned distribution, then calculate spread.
Common mistakes
Candidates often keep the old mean or assume conditioning always reduces variance. Conditioning can reduce, increase, or reshape spread depending on the information.
Practice the pattern
Use the LeetQuidity curriculum and calibration to turn this topic into a focused practice plan.