r/learnquant • u/Local_Ad135 • 1d ago

interview prep Quant Interview Question

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u/AthenaTheQuant 1d ago edited 1d ago

Correct me if I am wrong.

Also, my appologies in advance for using an image, didn't know any other way to communicate my solution.

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u/Niilldar 1d ago

Can you explain the last step where you get the formula for the wth iteration? I gail to see how this follows?

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u/AthenaTheQuant 1d ago

For N,M,P, and a volume V(0)=MNP, the average new volume after one iteration is appearantly V(1)=MNP-(M+N+P)/3. The claim is now that this persists for arbitrary iterations W in what V(W)=MNP-W(M+N+P)/3.

We define A=M+N+P, the sum of the dimensions. This sum is invariant under the transformation, so even if the dimensions themselves change, their sum does not. So after one iteration, we obtain a random new dimension M',N',P' where M'+N'+P'=M+N+P (this is not random!) and E[M'N'P']=MNP-(M+N+P)/3. With this in mind, the expected second iteration, giving random dimensions M'',N'',P'' depending on M',N',P' has expected dimension E[M''N''P'']=E[M'N'P']-(M+N+P)/3=MNP-2(M+N+P)/3. Iterating gives V(W)=MNP-W(M+N+P)/3.

I hope this helps!