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\nCorrelation between marginal distributions of a joint probability distribution
\nLet $F,G \\in \\mathcal{D}_{\\mathbb{R}^d}$ be two joint probability distributions with marginals $\\mu, \nu \\in \\mathcal{P}$. Show that\n\\begin{equation}\n\\rho = \\frac{\\int \\int f(x) g(y) d\\mu(x) d\nu(y)}{\\int \\int f(x) dx d\\mu(x) \\int g(y) dy d\nu(y)}\n\\end{equation}\nis a correlation function.\nMy attempt: Letting $\\mu(U), \nu(U)$ be the marginal distributions of two random variables $U,V \\in \\mathcal{U}$ we find for every $T,U \\in \\mathcal{U}$ that\n\\begin{equation}\n\\int T(U,y) d\nu(y) = \\int \\int T(x,y) d\nu(y) dx\n\\end{equation}\nand\n\\begin{equation}\n\\int T(x,U) d\\mu(x) = \\int \\int T(x,y) d\\mu(x) dy\n\\end{equation}\nhence\n\\begin{equation}\n\\int \\int T(x,y) d\\mu(x)\nhttps://documenter.getpostman.com/view/21879982/UzdtWnud https://documenter.getpostman.com/view/21889843/UzdtWnyw https://documenter.getpostman.com/view/21827949/UzdtWnyx https://documenter.getpostman.com/view/21911008/UzdtWnyu https://documenter.getpostman.com/view/21883157/UzdtWnue # Introduction\nWhat does your API do?
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