Construct the correlation matrix ((1,x,x),(1,x,1),(x,1,1)) = C. For any vector v, we need dot(v, dot(C,v)) >= 0. Choose v= (1,1,1) to get x >= -1/2. To get the upper bound of 1, compute det (C) to get 2x^3 -3x^2 +1 >= 0. We know x=-1/2 is a zero, and we have 2 more. We can easily check that 1 is a zero. Thus x in [-1/2, 1].

from the requirement that the correlation matrix be positive semi-definite, we'd get correlation coefficient in [-1/2,1]

let x be the correlation (1 x x) —> the correlation matrix A = (x 1 x) should be positive semi-definite (x x 1) —> |x| == 0 —> -1/2 = why ? because intuitively, if x1 and x2 are negatively correlated, x2 and x3 are negatively correlated, then my naivety tends to tell me that x1 and x3 move in the same direction, i.e. positively correlated. but it seems like negative values for x are also possible

let x be the correlation (1 x x) —> the correlation matrix A = (x 1 x) should be positive semi-definite (x x 1) —> |x| = 0 —> -1/2 why ? because intuitively, if x1 and x2 are negatively correlated, x2 and x3 are negatively correlated, then my naivety tends to tell me that x1 and x3 move in the same direction, i.e. positively correlated. but it seems like negative values for x are also possible

0, 1.