Now I have an arbitrary group of events $X_1,X_2,\ldots,X_m$(with no independence or correlation assumptions, nor distribution knowledge), and define a symmetric matrix $\mathbf{K}$ as below: $$ \mathbf{K}_{ij} = \text{Pr}[X_i\cap X_j]-\text{Pr}[X_i]\text{Pr}[X_j] $$ with $\text{Pr}[X_i]$ representing the probability associated with event $X_i$. Is $\mathbf{K}$ positive semi-definite?


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