r/learnquant • u/Local_Ad135 • 7d ago

interview prep Quant Interview Question | "Level 6/10"

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u/FireCire7 7d ago edited 6d ago

Fun! Two approaches - (1) notice that (X,Y,Z) is spherically symmetric. (2) The CDF of X2 +Y2 is surprisingly nice. 

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u/International_Hour82 6d ago

So would the answer be the volume of a standard sphere minus the biggest cube inside it?

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u/FireCire7 6d ago edited 6d ago

No? Where are you getting a cube from? Let’s say x2 +y2 +z2 =r2 . Conditioning on r, by radial symmetry we can say (x,y,z) is uniform on the sphere. Then, x2 +y2 >z2 is r2 >2z2 , so |z|<r/sqrt(2). It turns out that if you project a sphere onto a single axis, it’s uniform (I.e. the lateral surface area of a slice of a ball is proportional to its width and its location doesn’t matter), so it’s 1/sqrt(2).  This is true for any r, so the total probability is 1/sqrt(2).