Change Variables / Normalize
Rescale or transform the problem into its simplest equivalent form.
What is it?
Replace given quantities with normalized ones (length 1, mean 0, unit rate) or new variables that remove a constraint, solve, then convert back.
Why is it useful?
Many interview questions carry arbitrary units; normalizing reveals the structure and reduces arithmetic.
When should you think of it?
Think of it when the answer should scale with a parameter, or a substitution removes a constraint.
Think of this method when…
- A length , a rate or a scale that should factor out
- Uniform on instead of
- Normal with general mean and variance
- A substitution like that removes a lower bound
Core idea
Solve the unit-scale problem and multiply by the scale; or standardize and use the known distribution.
Step-by-step playbook
- 1Identify the scale or location parameter.
- 2Transform to the standard form.
- 3Solve the simpler problem.
- 4Transform the answer back.
Worked example
Worked example
are independent and uniform on . What is ?
- 1
Normalize: , with .
- 2
, then scale back.
- Answer:
Common mistakes
- Forgetting to convert the answer back.
- Scaling probabilities (they are unit-free).
Would you use this method?
. To find , should you standardize?
Practice this method
1 QuantSolve questions are tagged with this method.
Practice this method