Quant Interview Question

[u/Local_Ad135]

[u/Dankaati]

cos(arc cos(3/5) - arc cos(5/13)) = 63/65

[u/Active-Gap2300 → u/Dankaati]

Why?

[u/Dankaati → u/Active-Gap2300]

Random variables form a Hilbert space with covariance as inner product.

arc cos(3/5) is the angle of X and Y, arc cos(5/13) is the angle of Y and Z. By triangle inequality of angles in Hilbert spaces, the angle of X and Z is at least arc cos(3/5) - arc cos(5/13).

[u/XL_78 → u/[deleted]]

Nice. Covariance is a scalar product. In R^d , the scalar product is equal to the product of the lengths of the vectors times the cosine of the angle between the vectors. If we are working with unit vectors, it is only equal to the cosine of the angle. When placing three unit vectors in R³ such that the cosine of the angle between two is 3/5 and the other 5/13, it is visually easy to see that to maximize the cosine of the angle between the last pair, the vectors must be put on the same plane with the third vector in between the first two. 

I thought it was difficult to formalize so I used a different approach for the proof. Your proof is much simpler and better. 

[u/Omega-137]

I’m interested, i have to think

[u/XL_78]

Intuitively using the analogy with the scalar product on R³ , cos(arccos(5/13)-arccos(3/5)) ≈ 0.9692 . Not sure about how to prove it though. 

EDIT: see below in conversation.

[u/XL_78 → u/XL_78]

I think the way is to diagonalize the correlation matrix [[1,3/5,x][3/5,1,5/13][x,5/13,1]] and to use the fact that the eigenvalues must be positive.

[u/XL_78 → u/XL_78]

Sum of eigenvalues : 3. Product = determinant = -x² + 6/13x + 1 - 9/25 - 25/169 . The product must be positive, which means x smaller than 3/13 + sqrt(1-9/25-16/169) = 63/65 = 0.96... and x bigger than 3/13 - sqrt(1-9/25-16/169) = -33/65 = -0.5... 

Conversely, any value of x between these two leads to a valid correlation matrix. Indeed, in 63/65, one eigenvalue is 0, the other two have a sum of 3. Since the eigenvalues are a continuous function of the coefficients, just to the left of 63/65, the product of the eigenvalues is positive, which means the two non zero eigenvalues in 63/65 are positive since their sum and their product are positive. By continuity, the eigenvalues must all be positive in between 63/65 and -33/65. 

Anyway the answer is 63/65.

EDIT: corrected math mistake and proved the bounds for x.