As Liu Wen from Hebei University of Technology in Tianjin,

Chapter , Problem 29

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As Liu Wen from Hebei University of Technology in Tianjin, China, has noted in the April 2001 issue of The American Mathematical Monthly, in some reputable probability and statistics texts it has been asserted that if a two-dimensional distribution function F (x, y) has a continuous density of f (x, y), then f (x, y) = 2F (x, y) x y . (8.8) Furthermore, some intermediate textbooks in probability and statistics even assert that at a point of continuity for f (x, y), F (x, y) is twice differentiable, and (8.8) holds at that point. Let the joint probability density function of random variables X and Y be given by f (x, y) = 1 + 2xey if y 0, (1/2)ey x 0 1 2xey if y 0, 0 x (1/2)ey 1 + 2xey if y 0, (1/2)ey x 0 1 2xey if y 0, 0 x (1/2)ey 0 otherwise. Show that even though f is continuous everywhere, the partial derivatives of its distribution function F do not exist at (0, 0). This counterexample is constructed based on a general example given by Liu Wen in the aforementioned paper.