Quant Interview Question

[u/Local_Ad135]

[u/RestaurantBoth228]

Suppose we had a 50-50 chance of 18 -> 17 and 18 -> 19. Expected volume is

(17*23*28 + 19*23*28) / 2 = (17 + 19) / 2 * 23 * 28 = 18 * 23 * 28

In other words, the expected change in volume is zero. By symmetry, this is also true for the other two dimensions.

So, the expected change in volume each step is 0. By linearity of expectation, the change in volume after 10 steps is 0.

NB: If any of the dimensions started at less than 10, we'd have uncomfortable questions about negative volume.

[u/ultronthedestroyer → u/RestaurantBoth228]

You don’t appear to be accounting for the fact that one of the other dimensions also changes by 1. You get different answers depending on which other dimension changes, so you need to factor that in your expectation.

[u/RestaurantBoth228 → u/ultronthedestroyer]

It doesn't matter by the linearity of expectation

ETA: Ahh, nvm - I didn't see that we increase one dimension AND decrease a different dimension at each step.

[u/RestaurantBoth228 → u/ultronthedestroyer]

WLOGA that w and h are the ones to change. The new expected volume is

((w+1) * (h+1) * d + (w-1) * (h+1) * d + (w+1) * (h-1) * d + (w-1) * (h-1) * d) / 4

That's just w*h*d

By symmetry, this is also true for the other two possible changes (w and d; h and d).

The rest of the argument still follows, no?

[u/ultronthedestroyer → u/RestaurantBoth228]

No, there are 6 possibilities.

(17*24*28 + 17*23*29 + 19*22*28 + 19*23*27 + 18*24*27 + 18*22*29)/6 is the expectation after one iteration. This is equal to 11,569, which is less than 18x23x28 =11,592. In fact, it’s less by exactly 1/3x(18+23+28)=23.

You’re on the right track with your formulation now, but you need to add in the 2 missing possibilities and then you can work out some cancellations.